4.6: Generating Functions
As usual, our starting point is a random experiment modeled by a probability sace \((\Omega, \mathscr F, \P)\). A generating function of a real-valued random variable is an expected value of a certain transformation of the random variable involving another (deterministic) variable. Most generating functions share four important properties:
- Under mild conditions, the generating function completely determines the distribution of the random variable.
- The generating function of a sum of independent variables is the product of the generating functions
- The moments of the random variable can be obtained from the derivatives of the generating function.
- Ordinary (pointwise) convergence of a sequence of generating functions corresponds to the special convergence of the corresponding distributions.
Property 1 is perhaps the most important. Often a random variable is shown to have a certain distribution by showing that the generating function has a certain form. The process of recovering the distribution from the generating function is known as inversion . Property 2 is frequently used to determine the distribution of a sum of independent variables. By contrast, recall that the probability density function of a sum of independent variables is the convolution of the individual density functions, a much more complicated operation. Property 3 is useful because often computing moments from the generating function is easier than computing the moments directly from the probability density function. The last property is known as the continuity theorem . Often it is easer to show the convergence of the generating functions than to prove convergence of the distributions directly.
The numerical value of the generating function at a particular value of the free variable is of no interest, and so generating functions can seem rather unintuitive at first. But the important point is that the generating function as a whole encodes all of the information in the probability distribution in a very useful way. Generating functions are important and valuable tools in probability, as they are in other areas of mathematics, from combinatorics to differential equations.
We will study the three generating functions in the list below, which correspond to increasing levels of generality. The fist is the most restrictive, but also by far the simplest, since the theory reduces to basic facts about power series that you will remember from calculus. The third is the most general and the one for which the theory is most complete and elegant, but it also requires basic knowledge of complex analysis. The one in the middle is perhaps the one most commonly used, and suffices for most distributions in applied probability.
We will also study the characteristic function for multivariate distributions, although analogous results hold for the other two types. In the basic theory below, be sure to try the proofs yourself before reading the ones in the text.
Basic Theory
The Probability Generating Function
For our first generating function, assume that \(N\) is a random variable taking values in \(\N\).
The probability generating function \(P\) of \(N\) is defined by \[ P(t) = \E\left(t^N\right) \] for all \(t \in \R\) for which the expected value exists in \( \R \).
That is, \( P(t) \) is defined when \( \E\left(|t|^N\right) \lt \infty \). The probability generating function can be written nicely in terms of the probability density function.
Suppose that \(N\) has probability density function \(f\) and probability generating function \(P\). Then \[ P(t) = \sum_{n=0}^\infty f(n) t^n, \quad t \in (-r, r) \] where \( r \in [1, \infty] \) is the radius of convergence of the series.
Proof
The expansion follows from the discrete change of variables theorem for expected value. Note that the series is a power series in \( t \), and hence by basic calculus, converges absolutely for \( t \in (-r, r) \) where \( r \in [0, \infty] \) is the radius of convergence. But since \( \sum_{n=0}^\infty f(n) = 1 \) we must have \( r \ge 1 \), and the series converges absolutely at least for \( t \in [-1, 1] \).
In the language of combinatorics, \(P\) is the ordinary generating function of \(f\). Of course, if \( N \) just takes a finite set of values in \( \N \) then \( r = \infty \). Recall from calculus that a power series can be differentiated term by term, just like a polynomial. Each derivative series has the same radius of convergence as the original series (but may behave differently at the endpoints of the interval of convergence). We denote the derivative of order \(n\) by \(P^{(n)}\). Recall also that if \(n \in \N\) and \(k \in \N\) with \(k \le n\), then the number of permutations of size \(k\) chosen from a population of \(n\) objects is \[ n^{(k)} = n (n - 1) \cdots (n - k + 1) \] The following theorem is the inversion result for probability generating functions: the generating function completely determines the distribution.
Suppose again that \(N\) has probability density function \(f\) and probability generating function \(P\). Then \[ f(k) = \frac{P^{(k)}(0)}{k!}, \quad k \in \N \]
Proof
This is a standard result from the theory of power series. Differentiating \( k \) times gives \( P^{(k)}(t) = \sum_{n=k}^\infty n^{(k)} f(n) t^{n-k} \) for \( t \in (-r, r) \). Hence \( P^{(k)}(0) = k^{(k)} f(k) = k! f(k) \)
Our next result is not particularly important, but has a certain curiosity.
\(\P(N \text{ is even}) = \frac{1}{2}\left[1 + P(-1)\right]\).
Proof
Note that \[ P(1) + P(-1) = \sum_{n=0}^\infty f(n) + \sum_{n=0}^\infty (-1)^n f(n) = 2 \sum_{k=0}^\infty f(2 k) = 2 \P(N \text{ is even }) \] We can combine the two sum since we know that the series converge absolutely at 1 and \(-1\).
Recall that the factorial moment of \( N \) of order \( k \in \N \) is \( \E\left[N^{(k)}\right] \). The factorial moments can be computed from the derivatives of the probability generating function. The factorial moments, in turn, determine the ordinary moments about 0 (sometimes referred to as raw moments ).
Suppose that the radius of convergence \(r \gt 1\). Then \(P^{(k)}(1) = \E\left[N^{(k)}\right]\) for \(k \in \N\). In particular, \(N\) has finite moments of all orders.
Proof
As before, \( P^{(k)}(t) = \sum_{n=k}^\infty n^{(k)} f(n) t^{n-k} \) for \( t \in (-r, r) \). Hence if \( r \gt 1 \) then \( P^{(k)}(1) = \sum_{n=k}^\infty n^{(k)} f(n) = \E\left[N^{(k)}\right] \)
Suppose again that \( r \gt 1 \). Then
- \(\E(N) = P^\prime(1)\)
- \(\var(N) = P^{\prime \prime}(1) + P^\prime(1)\left[1 - P^\prime(1)\right]\)
Proof
- \( \E(N) = \E\left[N^{(1)}\right] = P^\prime(1) \).
- \( \E\left(N^2\right) = \E[N (N - 1)] + \E(N) = \E\left[N^{(2)}\right] + \E(N) = P^{\prime\prime}(1) + P^\prime(1) \). Hence from (a), \( \var(N) = P^{\prime\prime}(1) + P^\prime(1) - \left[P^\prime(1)\right]^2 \).
Suppose that \(N_1\) and \(N_2\) are independent random variables taking values in \(\N\), with probability generating functions \(P_1\) and \(P_2\) having radii of convergence \( r_1 \) and \( r_2 \), respectively. Then the probability generating function \( P \) of \(N_1 + N_2\) is given by \(P(t) = P_1(t) P_2(t)\) for \( \left|t\right| \lt r_1 \wedge r_2 \).
Proof
Recall that the expected product of independent variables is the product of the expected values. Hence \[ P(t) = \E\left(t^{N_1 + N_2}\right) = \E\left(t^{N_1} t^{N_2}\right) = \E\left(t^{N_1}\right) \E\left(t^{N_2}\right) = P_1(t) P_2(t), \quad \left|t\right| \lt r_1 \wedge r_2 \]
The Moment Generating Function
Our next generating function is defined more generally, so in this discussion we assume that the random variables are real-valued.
The moment generating function of \(X\) is the function \(M\) defined by \[ M(t) = \E\left(e^{tX}\right), \quad t \in \R \]
Note that since \(e^{t X} \ge 0\) with probability 1, \(M(t)\) exists, as a real number or \(\infty\), for any \(t \in \R\). But as we will see, our interest will be in the domain where \( M(t) \lt \infty \).
Suppose that \(X\) has a continuous distribution on \(\R\) with probability density function \(f\). Then \[ M(t) = \int_{-\infty}^\infty e^{t x} f(x) \, dx \]
Proof
This follows from the change of variables theorem for expected value.
Thus, the moment generating function of \(X\) is closely related to the Laplace transform of the probability density function \(f\). The Laplace transform is named for Pierre Simon Laplace, and is widely used in many areas of applied mathematics, particularly differential equations. The basic inversion theorem for moment generating functions (similar to the inversion theorem for Laplace transforms) states that if \(M(t) \lt \infty\) for \(t\) in an open interval about 0, then \(M\) completely determines the distribution of \(X\). Thus, if two distributions on \(\R\) have moment generating functions that are equal (and finite) in an open interval about 0, then the distributions are the same.
Suppose that \(X\) has moment generating function \(M\) that is finite in an open interval \( I \) about 0. Then \(X\) has moments of all orders and \[ M(t) = \sum_{n=0}^\infty \frac{\E\left(X^n\right)}{n!} t^n, \quad t \in I \]
Proof
Under the hypotheses, the expected value operator can be interchanged with the infinite series for the exponential function: \[ M(t) = \E\left(e^{t X}\right) = \E\left(\sum_{n=0}^\infty \frac{X^n}{n!} t^n\right) = \sum_{n=0}^\infty \frac{\E(X^n)}{n!} t^n, \quad t \in I \] The interchange is a special case of Fubini's theorem , named for Guido Fubini. For more details see the advanced section on properties of the integral in the chapter on Distributions.
So under the finite assumption in the last theorem , the moment generating function, like the probability generating function, is a power series in \( t \).
Suppose again that \( X \) has moment generating function \( M \) that is finite in an open interval about 0. Then \(M^{(n)}(0) = \E\left(X^n\right)\) for \(n \in \N\)
Proof
This follows by the same argument as above for the PGF : \( M^{(n)}(0) / n! \) is the coefficient of order \( n \) in the power series above, namely \( \E\left(X^n\right) / n! \). Hence \( M^{(n)}(0) = \E\left(X^n\right) \).
Thus, the derivatives of the moment generating function at 0 determine the moments of the variable (hence the name). In the language of combinatorics, the moment generating function is the exponential generating function of the sequence of moments. Thus, a random variable that does not have finite moments of all orders cannot have a finite moment generating function. Even when a random variable does have moments of all orders, the moment generating function may not exist. A counterexample is constructed below.
For nonnegative random variables (which are very common in applications), the domain where the moment generating function is finite is easy to understand.
Suppose that \( X \) takes values in \( [0, \infty) \) and has moment generating function \( M \). If \( M(t) \lt \infty \) for \( t \in \R \) then \( M(s) \lt \infty \) for \( s \le t \).
Proof
Since \( X \ge 0 \), if \( s \le t \) then \( s X \le t X \) and hence \( e^{s X} \le e^{t X} \). Hence \( \E\left(e^{s X}\right) \le \E\left(e^{t X} \right) \).
So for a nonnegative random variable, either \( M(t) \lt \infty \) for all \( t \in \R \) or there exists \( r \in (0, \infty) \) such that \( M(t) \lt \infty \) for \( t \lt r \). Of course, there are complementary results for non-positive random variables, but such variables are much less common. Next we consider what happens to the moment generating function under some simple transformations of the random variables.
Suppose that \(X\) has moment generating function \(M\) and that \(a, \, b \in \R\). The moment generating function \( N \) of \(Y = a + b X\) is given by \(N(t) = e^{a t} M(b t)\) for \( t \in \R \).
Proof
\( \E\left[e^{t (a + b X)}\right] = \E\left(e^{t a} e^{t b X}\right) = e^{t a} \E\left[e^{(t b) X}\right] = e^{a t} M(b t) \) for \(t \in \R).
Recall that if \( a \in \R \) and \( b \in (0, \infty) \) then the transformation \( a + b X \) is a location-scale transformation on the distribution of \( X \), with location parameter \(a\) and scale parameter \(b\). Location-scale transformations frequently arise when units are changed, such as length changed from inches to centimeters or temperature from degrees Fahrenheit to degrees Celsius.
Suppose that \(X_1\) and \(X_2\) are independent random variables with moment generating functions \(M_1\) and \(M_2\) respectively. The moment generating function \( M \) of \(Y = X_1 + X_2\) is given by \(M(t) = M_1(t) M_2(t)\) for \( t \in \R \).
Proof
As with the PGF, the proof for the MGF relies on the law of exponents and the fact that the expected value of a product of independent variables is the product of the expected values: \[ \E\left[e^{t (X_1 + X_2)}\right] = \E\left(e^{t X_1} e^{t X_2}\right) = \E\left(e^{t X_1}\right) \E\left(e^{t X_2}\right) = M_1(t) M_2(t), \quad t \in \R \]
The probability generating function of a variable can easily be converted into the moment generating function of the variable.
Suppose that \(X\) is a random variable taking values in \(\N\) with probability generating function \(G\) having radius of convergence \( r \). The moment generating function \( M \) of \(X\) is given by \(M(t) = G\left(e^t\right)\) for \( t \lt \ln(r) \).
Proof
\( M(t) = \E\left(e^{t X}\right) = \E\left[\left(e^t\right)^X\right] = G\left(e^t\right) \) for \( e^t \lt r \).
The following theorem gives the Chernoff bounds , named for the mathematician Herman Chernoff. These are upper bounds on the tail events of a random variable.
If \(X\) has moment generating function \(M\) then
- \(\P(X \ge x) \le e^{-t x} M(t)\) for \(t \gt 0\)
- \(\P(X \le x) \le e^{-t x} M(t)\) for \(t \lt 0\)
Proof
- From Markov's inequality, \(\P(X \ge x) = \P\left(e^{t X} \ge e^{t x}\right) \le \E\left(e^{t X}\right) \big/ e^{t x} = e^{-t x} M(t) \) if \(t \gt 0\).
- Similarly, \(\P(X \le x) = \P\left(e^{t X} \ge e^{t x}\right) \le e^{-t x} M(t) \) if \(t \lt 0\).
Naturally, the best Chernoff bound (in either (a) or (b)) is obtained by finding \(t\) that minimizes \(e^{-t x} M(t)\).
The Characteristic Function
Our last generating function is the nicest from a mathematical point of view. Once again, we assume that our random variables are real-valued.
The characteristic function of \(X\) is the function \( \chi \) defined by by \[ \chi(t) = \E\left(e^{i t X}\right) = \E\left[\cos(t X)\right] + i \E\left[\sin(t X)\right], \quad t \in \R \]
Note that \(\chi\) is a complex valued function, and so this subsection requires some basic knowledge of complex analysis. The function \(\chi\) is defined for all \(t \in \R\) because the random variable in the expected value is bounded in magnitude. Indeed, \(\left|e^{i t X}\right| = 1\) for all \(t \in \R\). Many of the properties of the characteristic function are more elegant than the corresponding properties of the probability or moment generating functions, because the characteristic function always exists.
If \(X\) has a continuous distribution on \(\R\) with probability density function \(f\) and characteristic function \( \chi \) then \[ \chi(t) = \int_{-\infty}^{\infty} e^{i t x} f(x) dx, \quad t \in \R \]
Proof
This follows from the change of variables theorem for expected value, albeit a complex version.
Thus, the characteristic function of \(X\) is closely related to the Fourier transform of the probability density function \(f\). The Fourier transform is named for Joseph Fourier, and is widely used in many areas of applied mathematics.
As with other generating functions, the characteristic function completely determines the distribution. That is, random variables \(X\) and \(Y\) have the same distribution if and only if they have the same characteristic function. Indeed, the general inversion formula given next is a formula for computing certain combinations of probabilities from the characteristic function.
Suppose again that \( X \) has characteristic function \( \chi \). If \( a, \, b \in \R \) and \(a \lt b\) then \[ \int_{-n}^n \frac{e^{-i a t} - e^{- i b t}}{2 \pi i t} \chi(t) \, dt \to \P(a \lt X \lt b) + \frac{1}{2}\left[\P(X = b) - \P(X = a)\right] \text{ as } n \to \infty \]
The probability combinations on the right side completely determine the distribution of \(X\). A special inversion formula holds for continuous distributions:
Suppose that \(X\) has a continuous distribution with probability density function \(f\) and characteristic function \( \chi \). At every point \(x \in \R\) where \(f\) is differentiable, \[ f(x) = \frac{1}{2 \pi} \int_{-\infty}^\infty e^{-i t x} \chi(t) \, dt \]
This formula is essentially the inverse Fourrier transform . As with the other generating functions, the characteristic function can be used to find the moments of \(X\). Moreover, this can be done even when only some of the moments exist.
Suppose again that \( X \) has characteristic function \( \chi \). If \(n \in \N_+\) and \(\E\left(\left|X^n\right|\right) \lt \infty\). Then \[ \chi(t) = \sum_{k=0}^n \frac{\E\left(X^k\right)}{k!} (i t)^k + o(t^n) \] and therefore \(\chi^{(n)}(0) = i^n \E\left(X^n\right)\).
Details
Recall that the last term is a generic function that satisfies \(o(t^n) / t^n \to 0\) as \(t \to \infty\).
Next we consider how the characteristic function is changed under some simple transformations of the variables.
Suppose that \(X\) has characteristic function \(\chi\) and that \(a, \, b \in \R\). The characteristic function \( \psi \) of \(Y = a + b X\) is given by \(\psi(t) = e^{i a t} \chi(b t)\) for \( t \in \R \).
Proof
The proof is just like the one for the MGF: \( \psi(t) = \E\left[e^{i t (a + b X)}\right] = \E\left(e^{i t a} e^{i t b X}\right) = e^{i t a} \E\left[e^{i (t b) X}\right] = e^{i a t} \chi(b t) \) for \(t \in \R\).
Suppose that \(X_1\) and \(X_2\) are independent random variables with characteristic functions \(\chi_1\) and \(\chi_2\) respectively. The characteristic function \( \chi \) of \(Y = X_1 + X_2\) is given by \(\chi(t) = \chi_1(t) \chi_2(t)\) for \( t \in \R \).
Proof
Again, the proof is just like the one for the MGF: \[ \chi(t) = \E\left[e^{i t (X_1 + X_2)}\right] = \E\left(e^{i t X_1} e^{i t X_2}\right) = \E\left(e^{i t X_1}\right) \E\left(e^{i t X_2}\right) = \chi_1(t) \chi_2(t), \quad t \in \R \]
The characteristic function of a random variable can be obtained from the moment generating function, under the basic existence condition that we saw earlier.
Suppose that \(X\) has moment generating function \(M\) that satisfies \(M(t) \lt \infty\) for \(t\) in an open interval \(I\) about 0. Then the characteristic function \(\chi\) of \(X\) satisfies \(\chi(t) = M(i t)\) for \(t \in I\).
The final important property of characteristic functions that we will discuss relates to convergence in distribution. Suppose that \((X_1, X_2, \ldots)\) is a sequence of real-valued random with characteristic functions \((\chi_1, \chi_2, \ldots)\) respectively. Since we are only concerned with distributions, the random variables need not be defined on the same probability space.
The Continuity Theorem
- If the distribution of \(X_n\) converges to the distribution of a random variable \(X\) as \(n \to \infty\) and \(X\) has characteristic function \(\chi\), then \(\chi_n(t) \to \chi(t)\) as \(n \to \infty\) for all \(t \in \R\).
- Conversely, if \(\chi_n(t)\) converges to a function \(\chi(t)\) as \(n \to \infty\) for \(t\) in an open interval about 0, and if \(\chi\) is continuous at 0, then \(\chi\) is the characteristic function of a random variable \(X\), and the distribution of \(X_n\) converges to the distribution of \(X\) as \(n \to \infty\).
There are analogous versions of the continuity theorem for probability generating functions and moment generating functions. The continuity theorem can be used to prove the central limit theorem, one of the fundamental theorems of probability. Also, the continuity theorem has a straightforward generalization to distributions on \(\R^n\).
The Joint Characteristic Function
All of the generating functions that we have discussed have multivariate extensions. However, we will discuss the extension only for the characteristic function, the most important and versatile of the generating functions. There are analogous results for the other generating functions. So in this discussion, we assume that \((X, Y)\) is a random vector for our experiment, taking values in \(\R^2\).
The (joint) characteristic function \( \chi \) of \((X, Y)\) is defined by \[ \chi(s, t) = \E\left[\exp(i s X + i t Y)\right], \quad (s, t) \in \R^2 \]
Once again, the most important fact is that \(\chi\) completely determines the distribution: two random vectors taking values in \(\R^2\) have the same characteristic function if and only if they have the same distribution.
The joint moments can be obtained from the derivatives of the characteristic function.
Suppose that \( (X, Y) \) has characteristic function \( \chi \). If \(m, \, n \in \N\) and \(\E\left(\left|X^m Y^n\right|\right) \lt \infty\) then \[ \chi^{(m, n)}(0, 0) = e^{i \, (m + n)} \E\left(X^m Y^n\right) \]
The marginal characteristic functions and the characteristic function of the sum can be easily obtained from the joint characteristic function:
Suppose again that \( (X, Y) \) has characteristic function \( \chi \), and let \(\chi_1\), \(\chi_2\), and \(\chi_+\) denote the characteristic functions of \(X\), \(Y\), and \(X + Y\), respectively. For \(t \in \R\)
- \(\chi(t, 0) = \chi_1(t)\)
- \(\chi(0, t) = \chi_2(t)\)
- \(\chi(t, t) = \chi_+(t)\)
Proof
All three results follow immediately from the definitions.
Suppose again that \( \chi_1 \), \( \chi_2 \), and \( \chi \) are the characteristic functions of \(X\), \(Y\), and \( (X, Y) \) respectively. Then \( X \) and \( Y \) are independent if and only if \(\chi(s, t) = \chi_1(s) \chi_2(t)\) for all \((s, t) \in \R^2\).
Naturally, the results for bivariate characteristic functions have analogies in the general multivariate case. Only the notation is more complicated.
Examples and Applications
As always, be sure to try the computational problems yourself before expanding the solutions and answers in the text.
Dice
Recall that an ace-six flat die is a six-sided die for which faces numbered 1 and 6 have probability \( \frac{1}{4} \) each, while faces numbered 2, 3, 4, and 5 have probability \( \frac{1}{8} \) each. Similarly, a 3-4 flat die is a six-sided die for which faces numbered 3 and 4 have probability \( \frac{1}{4} \) each, while faces numbered 1, 2, 5, and 6 have probability \( \frac{1}{8} \) each.
 Use probability generating functions to find the probability density function of the sum of the scores.
 Then \( X \) and \( Y \) have PGFs \( P \) and \( Q \) given by \begin{align*} P(t) &= \frac{1}{4} t + \frac{1}{8} t^2 + \frac{1}{8} t^3 + \frac{1}{8} t^4 + \frac{1}{8} t^5 + \frac{1}{4} t^6, \quad t \in \R\\ Q(t) &= \frac{1}{8} t + \frac{1}{8} t^2 + \frac{1}{4} t^3 + \frac{1}{4} t^4 + \frac{1}{8} t^5 + \frac{1}{8} t^6, \quad t \in \R \end{align*} Hence \( X + Y \) has PGF \( P Q \). Expanding (a computer algebra program helps) gives \[ P(t) Q(t) = \frac{1}{32} t^2 + \frac{3}{64} t^3 + \frac{3}{32} t^4 + \frac{1}{8} t^5 + \frac{1}{8} t^6 + \frac{5}{32} t^7 + \frac{1}{8} t^8 + \frac{1}{8} t^9 + \frac{3}{32} t^{10} + \frac{3}{64} t^{11} + \frac{1}{32} t^{12}, \quad t \in \R\] Thus the PDF \( f \) of \( X + Y \) is given by \( f(2) = f(12) = \frac{1}{32} \), \( f(3) = f(11) = \frac{3}{64} \), \( f(4) = f(10) = \frac{3}{32} \), \( f(5) = f(6) = f(8) = f(9) = \frac{1}{8} \) and \( f(7) = \frac{5}{32} \).
Two fair, 6-sided dice are rolled. One has faces numbered \( (0, 1, 2, 3, 4, 5) \) and the other has faces numbered \( (0, 6, 12, 18, 24, 30) \). Use probability generating functions to find the probability density function of the sum of the scores, and identify the distribution.
 Then \( X \) and \( Y \) have PGFs \( P \) and \( Q \) given by \begin{align*} P(t) &= \frac{1}{6} \sum_{k=0}^5 t^k \quad t \in \R \\ Q(t) &= \frac{1}{6} \sum_{j=0}^5 t^{6 j} \quad t \in \R \end{align*} Hence \( X + Y \) has PGF \( P Q \). Simplifying gives \[ P(t) Q(t) = \frac{1}{36} \sum_{j=0}^5 \sum_{k=0}^5 t^{6 j + k} = \frac{1}{36} \sum_{n=0}^{35} t^n, \quad t \in \R\] Hence \( X + Y \) is uniformly distributed on \( \{0, 1, 2, \ldots, 35\} \).
Suppose that random variable \( Y \) has probability generating function \( P \) given by \[ P(t) = \left(\frac{2}{5} t + \frac{3}{10} t^2 + \frac{1}{5} t^3 + \frac{1}{10} t^4\right)^5, \quad t \in \R \]
- Interpret \( Y \) in terms of rolling dice.
- Use the probability generating function to find the first two factorial moments of \( Y \).
- Use (b) to find the variance of \( Y \).
 \( Y \) is the sum of the scores when the die is rolled 5 times.
- \( \E(Y) = P^\prime(1) = 10 \), \( \E[Y(Y - 1)] = P^{\prime \prime}(1) = 95 \)
- \( \var(Y) = 5 \)
Bernoulli Trials
Suppose \(X\) is an indicator random variable with \(p = \P(X = 1)\), where \(p \in [0, 1]\) is a parameter. Then \(X\) has probability generating function \(P(t) = 1 - p + p t\) for \(t \in \R\).
Proof
\( P(t) = \E\left(t^X\right) = t^0 (1 - p) + t^1 p = 1 - p + p t \) for \( t \in \R \).
Recall that a Bernoulli trials process is a sequence \((X_1, X_2, \ldots)\) of independent, identically distributed indicator random variables. In the usual language of reliability, \(X_i\) denotes the outcome of trial \(i\), where 1 denotes success and 0 denotes failure. The probability of success \(p = \P(X_i = 1)\) is the basic parameter of the process. The process is named for Jacob Bernoulli. A separate chapter on the Bernoulli Trials explores this process in more detail.
For \(n \in \N_+\), the number of successes in the first \(n\) trials is \(Y_n = \sum_{i=1}^n X_i\). Recall that this random variable has the binomial distribution with parameters \(n\) and \(p\), which has probability density function \( f_n \) given by \[ f_n(y) = \binom{n}{y} p^y (1 - p)^{n - y}, \quad y \in \{0, 1, \ldots, n\} \]
Random variable \(Y_n\) has probability generating function \(P_n\) given by \(P_n(t) = (1 - p + p t)^n\) for \( t \in \R \).
Proof
This follows immediately from the PGF of an indicator variable and the result for sums of independent variables .
Rando variable \(Y_n\) has the following parameters:
- \(\E\left[Y_n^{(k)}\right] = n^{(k)} p^k\)
- \(\E\left(Y_n\right) = n p\)
- \(\var\left(Y_n\right) = n p (1 - p)\)
- \(\P(Y_n \text{ is even}) = \frac{1}{2}\left[1 - (1 - 2 p)^n\right]\)
Proof
- Repeated differentiation gives \( P^{(k)}(t) = n^{(k)} p^k (1 - p + p t)^{n-k} \). Hence \( P^{(k)}(1) = n^{(k)} p^k \), which is \( \E\left[X^{(k)} \right]\) by the moment result above.
- This follows from the formula for mean .
- This follows from the formula for variance .
- This follows from the even value formula .
Suppose that \(U\) has the binomial distribution with parameters \(m \in \N_+\) and \(p \in [0, 1]\), \(V\) has the binomial distribution with parameters \(n \in \N_+\) and \(q \in [0, 1]\), and that \(U\) and \(V\) are independent.
- If \(p = q\) then \(U + V\) has the binomial distribution with parameters \(m + n\) and \(p\).
- If \(p \ne q\) then \(U + V\) does not have a binomial distribution.
Proof
From the result for sums of independent variables and the PGF of the binomial distribution , note that the probability generating function of \(U + V\) is \(P(t) = (1 - p + p t)^m (1 - q + q t)^n\) for \(t \in \R\).
- If \( p = q \) then \( U + V \) has PGF \( P(t) = (1 - p + p t)^{m + n} \), which is the PGF of the binomial distribution with parameters \( m + n \) and \( p \).
- On the other hand, if \( p \ne q \), the PGF \( P \) does not have the functional form of a binomial PGF.
Suppose now that \( p \in (0, 1] \). The trial number \( N \) of the first success in the sequence of Bernoulli trials has the geometric distribution on \( \N_+ \) with success parameter \( p \). The probability density function \( h \) is given by \[h(n) = p (1 - p)^{n-1}, \quad n \in \N_+\] The geometric distribution is studied in more detail in the chapter on Bernoulli trials.
Let \(Q\) denote the probability generating function of \(N\). Then
- \(Q(t) = \frac{p t}{1 - (1 - p)t}\) for \(-\frac{1}{1 - p} \lt t \lt \frac{1}{1 - p}\)
- \(\E\left[N^{(k)}\right] = k! \frac{(1 - p)^{k-1}}{p^k}\) for \( k \in \N \)
- \(\E(N) = \frac{1}{p}\)
- \(\var(N) = \frac{1 - p}{p^2}\)
- \(\P(N \text{ is even}) = \frac{1 - p}{2 - p}\)
Proof
- Using the formula for the sum of a geometric series, \[ Q(t) = \sum_{n=1}^\infty (1 - p)^{n-1} p t^n = p t \sum_{n=1}^\infty [(1 - p) t]^{n-1} = \frac{p t}{1 - (1 - p) t}, \quad \left|(1 - p) t\right| \lt 1 \]
- Repeated differentiation gives \( H^{(k)}(t) = k! p (1 - p)^{k-1} \left[1 - (1 - p) t\right]^{-(k+1)} \) and then the result follows from the inversion formula .
- This follows from (b) and the formula for mean .
- This follows from (b) and the formula for variance .
- This follows from even value formula .
The probability that \( N \) is even comes up in the alternating coin tossing game with two players.
The Poisson Distribution
Recall that the
Poisson distribution
has probability density function \(f\) given by \[ f(n) = e^{-a} \frac{a^n}{n!}, \quad n \in \N \] where \(a \in (0, \infty)\) is a parameter. The Poisson distribution is named after Simeon Poisson and is widely used to model the number of
random points
in a region of time or space; the parameter is proportional to the size of the region of time or space. The Poisson distribution is studied in more detail in the chapter on the Poisson Process.
Suppose that \(N\) has Poisson distribution with parameter \(a \in (0, \infty)\). Let \(P_a\) denote the probability generating function of \(N\). Then
- \(P_a(t) = e^{a (t - 1)}\) for \(t \in \R\)
- \(\E\left[N^{(k)}\right] = a^k\)
- \(\E(N) = a\)
- \(\var(N) = a\)
- \(\P(N \text{ is even}) = \frac{1}{2}\left(1 + e^{-2 a}\right)\)
Proof
- Using the exponential series, \[ P_a(t) = \sum_{n=0}^\infty e^{-a} \frac{a^n}{n!} t^n = e^{-a} \sum_{n=0}^\infty \frac{(a t)^n}{n!} = e^{-a} e^{a t}, \quad t \in \R \]
- Repeated differentiation gives \( P_a^{(k)}(t) = e^{a (t - 1)} a^k \), so the result follows from inversion formula .
- This follows from (b) and the formula for mean .
- This follows from (b) and the formula for variance .
- This follows from even value formula .
The Poisson family of distributions is closed with respect to sums of independent variables, a very important property.
Suppose that \(X, \, Y\) have Poisson distributions with parameters \(a, \, b \in (0, \infty)\), respectively, and that \(X\) and \(Y\) are independent. Then \(X + Y\) has the Poisson distribution with parameter \(a + b\).
Proof
In the notation of the previous result , note that \( P_a P_b = P_{a+b} \).
The right distribution function of the Poisson distribution does not have a simple, closed-form expression. The following exercise gives an upper bound.
Suppose that \(N\) has the Poisson distribution with parameter \(a \gt 0\). Then \[ \P(N \ge n) \le e^{n - a} \left(\frac{a}{n}\right)^n, \quad n \gt a \]
Proof
The PGF of \( N \) is \( P(t) = e^{a (t - 1)} \) and hence the MGF is \( P\left(e^t\right) = \exp\left(a e^t - a\right) \). From the Chernov bounds we have \[ \P(N \ge n) \le e^{-t n} \exp\left(a e^t - a\right) = \exp\left(a e^t - a - tn\right) \] If \( n \gt a \) the expression on the right is minimized when \( t = \ln(n / a) \). Substituting gives the upper bound.
The following theorem gives an important convergence result that is explored in more detail in the chapter on the Poisson process.
Suppose that \( p_n \in (0, 1) \) for \( n \in \N_+ \) and that \( n p_n \to a \in (0, \infty) \) as \( n \to \infty \). Then the binomial distribution with parameters \( n \) and \( p_n \) converges to the Poisson distribution with parameter \( a \) as \( n \to \infty \).
Proof
Let \(P_n\) denote the probability generating function of the binomial distribution with parameters \(n\) and \(p_n\). From the PGF of the binomial distribution we have \[ P_n(t) = \left[1 + p_n (t - 1)\right]^n = \left[1 + \frac{n p_n (t - 1)}{n}\right]^n, \quad t \in \R\] Using a famous theorem from calculus, \( P_n(t) \to e^{a (t - 1)} \) as \( n \to \infty \). But this is the PGF of the Poisson distribution with parameter \( a \), so the result follows from the continuity theorem for PGFs.
The Exponential Distribution
Recall that the exponential distribution is a continuous distribution on \([0, \infty)\) with probability density function \(f\) given by \[ f(t) = r e^{-r t}, \quad t \in (0, \infty) \] where \(r \in (0, \infty)\) is the rate parameter . This distribution is widely used to model failure times and other random times, and in particular governs the time between arrivals in the Poisson model. The exponential distribution is studied in more detail in the chapter on the Poisson Process.
Suppose that \(T\) has the exponential distribution with rate parameter \(r \in (0, \infty)\) and let \(M\) denote the moment generating function of \(T\). Then
- \(M(s) = \frac{r}{r - s}\) for \(s \in (-\infty, r)\).
- \(\E(T^n) = n! / r^n\) for \(n \in \N\)
Proof
- \( M(s) = \E\left(e^{s T}\right) = \int_0^\infty e^{s t} r e^{-r t} \, dt = \int_0^\infty r e^{(s - r) t} \, dt = \frac{r}{r - s} \) for \( s \lt r \).
- \( M^{(n)}(s) = \frac{r n!}{(r - s)^{n+1}} \) for \( n \in \N \)
Suppose that \((T_1, T_2, \ldots)\) is a sequence of independent random variables, each having the exponential distribution with rate parameter \(r \in (0, \infty)\). For \(n \in \N_+\), the moment generating function \( M_n \) of \(U_n = \sum_{i=1}^n T_i\) is given by \[ M_n(s) = \left(\frac{r}{r - s}\right)^n, \quad s \in (-\infty, r) \]
Proof
This follows from the previous result and the result for sums of independent variables .
Random variable \(U_n\) has the Erlang distribution with shape parameter \(n\) and rate parameter \(r\), named for Agner Erlang. This distribution governs the \( n \)th arrival time in the Poisson model. The Erlang distribution is a special case of the gamma distribution and is studied in more detail in the chapter on the Poisson Process.
Uniform Distributions
Suppose that \( a, \, b \in \R \) and \( a \lt b \). Recall that the continuous uniform distribution on the interval \( [a, b] \) has probability density function \( f \) given by \[ f(x) = \frac{1}{b - a}, \quad x \in [a, b] \] The distribution corresponds to selecting a point at random from the interval. Continuous uniform distributions arise in geometric probability and a variety of other applied problems.
Suppose that \(X\) is uniformly distributed on the interval \([a, b]\) and let \(M\) denote the moment generating function of \(X\). Then
- \(M(t) = \frac{e^{b t} - e^{a t}}{(b - a)t}\) if \( t \ne 0 \) and \( M(0) = 1 \)
- \(\E\left(X^n\right) = \frac{b^{n+1} - a^{n + 1}}{(n + 1)(b - a)}\) for \(n \in \N\)
Proof
- \( M(t) = \int_a^b e^{t x} \frac{1}{b - a} \, dx = \frac{e^{b t} - e^{a t}}{(b - a)t}\) if \( t \ne 0 \). Trivially \( M(0) = 1 \)
- This is a case where the MGF is not helpful, and it's much easier to compute the moments directly: \( \E\left(X^n\right) = \int_a^b x^n \frac{1}{b - a} \, dx = \frac{b^{n+1} - a^{n + 1}}{(n + 1)(b - a)} \)
Suppose that \((X, Y)\) is uniformly distributed on the triangle \(T = \{(x, y) \in \R^2: 0 \le x \le y \le 1\}\). Compute each of the following:
- The joint moment generating function of \((X, Y)\).
- The moment generating function of \(X\).
- The moment generating function of \(Y\).
- The moment generating function of \(X + Y\).
Answer
- \(M(s, t) = 2 \frac{e^{s+t} - 1}{s (s + t)} - 2 \frac{e^t - 1}{s t}\) if \( s \ne 0, \; t \ne 0\). \( M(0, 0) = 1 \)
- \(M_1(s) = 2 \left(\frac{e^2}{s^2} - \frac{1}{s^2} - \frac{1}{s}\right)\) if \(s \ne 0\). \( M_1(0) = 1 \)
- \(M_2(t) = 2 \frac{t e^t - e^t + 1}{t^2}\) if \( t \ne 0\). \( M_2(0) = 1 \)
- \(M_+(t) = \frac{e^{2 t} - 1}{t^2} - 2 \frac{e^t - 1}{t^2}\) if \(t \ne 0\). \( M_+(0) = 1 \)
A Bivariate Distribution
Suppose that \( (X, Y) \) has probability density function \(f\) given by \(f(x, y) = x + y\) for \((x, y) \in [0, 1]^2\). Compute each of the following:
- The joint moment generating function \( (X, Y) \).
- The moment generating function of \(X\).
- The moment generating function of \(Y\).
- The moment generating function of \(X + Y\).
Answer
- \(M(s, t) = \frac{e^{s+t}(-2 s t + s + t) + e^s(s t - s - t) + s + t}{s^2 t^2}\) if \(s \ne 0, \, t \ne 0\). \( M(0, 0) = 1 \)
- \(M_1(s) = \frac{3 s e^2 - 2 e^2 - s + 2}{2 s^2}\) if \(s \ne 0\). \( M_1(0) = 1 \)
- \(M_2(t) = \frac{3 t e^t - 2 e^t - t + 2}{2 t^2}\) if \(t \ne 0\). \( M_2(0) = 1 \)
- \(M_+(t) = \frac{[e^{2 t}(1 - t) + e^t (t - 2) + 1]}{t^3}\) if \(t \ne 0\). \( M_+(0) = 1 \)
The Normal Distribution
Recall that the standard normal distribution is a continuous distribution on \(\R) with probability density function \(\phi\) given by \[ \phi(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^2}, \quad z \in \R \] Normal distributions are widely used to model physical measurements subject to small, random errors and are studied in more detail in the chapter on Special Distributions.
Suppose that \(Z\) has the standard normal distribution and let \(M\) denote the moment generating function of \(Z\). Then
- \(M(t) = e^{\frac{1}{2} t^2}\) for \(t \in \R\)
- \(\E\left(Z^n\right) = 1 \cdot 3 \cdots (n - 1)\) if \( n \) is even and \(\E\left(Z^n\right) = 0\) if \(n\) is odd.
Proof
- First, \[ M(t) = \E\left(e^{t Z}\right) = \int_{-\infty}^\infty e^{t z} \frac{1}{\sqrt{2 \pi}} e^{-z^2 / 2} \, dz = \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi}} \exp\left(-\frac{z^2}{2} + t z\right) \, dz \] Completing the square in \( z \) gives \(\exp\left(-\frac{z^2}{2} + t z\right) = \exp\left[\frac{1}{2} t^2 - \frac{1}{2}(z - t)^2 \right] = e^{\frac{1}{2} t^2} \exp\left[-\frac{1}{2} (z - t)^2\right] \). hence \[ M(t) = e^{\frac{1}{2} t^2} \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi}} \exp\left[-\frac{1}{2} (z - t)^2\right] \, dz = e^{\frac{1}{2} t^2} \] because the function of \( z \) in the last integral is the probability density function for the normal distribution with mean \( t \) and variance 1.
- Note that \( M^\prime(t) = t M(t) \). Thus, repeated differentiation gives \( M^{(n)}(t) = p_n(t) M(t) \) for \( n \in \N \), where \( p_n \) is a polynomial of degree \( n \) satisfying \( p_{n+1}^\prime(t) = t p_n(t) + p_n^\prime(t) \). Since \( p_0 = 1 \), it's easy to see that \( p_n \) has only even or only odd terms, depending on whether \( n \) is even or odd, respectively. Thus, \( \E\left(X^n\right) = p_n(0) \). This is 0 if \( n \) is odd, and is the constant term \( 1 \cdot 3 \cdots (n - 1) \) if \( n \) is even. Of course, we can also see that the odd order moments must be 0 by symmetry.
More generally, for \(\mu \in \R\) and \(\sigma \in (0, \infty)\), recall that the normal distribution with mean \(\mu\) and standard deviation \(\sigma\) is a continuous distribution on \(\R\) with probability density function \( f \) given by \[ f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp\left[-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2\right], \quad x \in \R \] Moreover, if \( Z \) has the standard normal distribution, then \( X = \mu + \sigma Z \) has the normal distribution with mean \( \mu \) and standard deviation \( \sigma \). Thus, we can easily find the moment generating function of \( X \):
Suppose that \(X\) has the normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The moment generating function of \(X\) is \[M(t) = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right), \quad t \in \R\]
Proof
This follows easily the previous result and the result for linear transformations : \( X = \mu + \sigma Z \) where \( Z \) has the standard normal distribution. Hence \[ M(t) = \E\left(e^{t X}\right) = e^{\mu t} \E\left(e^{\sigma t Z}\right) = e^{\mu t} e^{\frac{1}{2} \sigma^2 t ^2}, \quad t \in \R \]
So the normal family of distributions in closed under location-scale transformations. The family is also closed with respect to sums of independent variables:
If \(X\) and \(Y\) are independent, normally distributed random variables then \(X + Y\) has a normal distribution.
Proof
Suppose that \( X \) has the normal distribution with mean \( \mu \in \R \) and standard deviation \( \sigma \in (0, \infty) \), and that \( Y \) has the normal distribution with mean \( \nu \in \R \) and standard deviation \( \tau \in (0, \infty) \). By (14) , the MGF of \( X + Y \) is \[ M_{X+Y}(t) = M_X(t) M_Y(t) = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right) \exp\left(\nu t + \frac{1}{2} \tau^2 t^2\right) = \exp\left[(\mu + \nu) t + \frac{1}{2}\left(\sigma^2 + \tau^2\right) t^2 \right] \] which we recognize as the MGF of the normal distribution with mean \( \mu + \nu \) and variance \( \sigma^2 + \tau^2 \). Of course, we already knew that \( \E(X + Y) = \E(X) + \E(Y) \), and since \( X \) and \( Y \) are independent, \( \var(X + Y) = \var(X) + \var(Y) \), so the new information is that the distribution is also normal.
The Pareto Distribution
Recall that the Pareto distribution is a continuous distribution on \([1, \infty) with probability density function \(f\) given by \[ f(x) = \frac{a}{x^{a + 1}}, \quad x \in [1, \infty) \] where \(a \in (0, \infty)\) is the shape parameter. The Pareto distribution is named for Vilfredo Pareto. It is a heavy-tailed distribution that is widely used to model financial variables such as income. The Pareto distribution is studied in more detail in the chapter on Special Distributions.
Suppose that \(X\) has the Pareto distribution with shape parameter \( a \), and let \( M \) denote the moment generating function of \( X \). Then
- \(\E\left(X^n\right) = \frac{a}{a - n}\) if \(n \lt a\) and \(\E\left(X^n\right) = \infty\) if \(n \ge a\)
- \(M(t) = \infty\) for \(t \gt 0\)
Proof
- We have seen this computation before. \( \E\left(X^n\right) = \int_1^\infty x^n \frac{a}{x^{a+1}} \, dx = \int_1^\infty x^{n - a - 1} \, dx \). The integral evaluates to \( \frac{a}{a - n} \) if \( n \lt a \) and \( \infty \) if \( n \ge a \).
- This follows from part (a). Since \( X \ge 1 \), \( M(t) \) is increasing in \( t \). Thus \( M(t) \le 1 \) if \( t \lt 0 \). If \( M(t) \lt \infty \) for some \( t \gt 0 \), then \( M(t) \) would be finite for \( t \) in an open interval about 0, in which case \( X \) would have finite moments of all orders. Of course, it's also easy to see directly from the integral that \( M(t) = \infty \) for \( t \gt 0 \)
On the other hand, like all distributions on \( \R \), the Pareto distribution has a characteristic function. However, the characteristic function of the Pareto distribution does not have a simple, closed form.
The Cauchy Distribution
Recall that the (standard) Cauchy distribution is a continuous distribution on \(\R\) with probability density function \(f\) given by \[ f(x) = \frac{1}{\pi \left(1 + x^2\right)}, \quad x \in \R \] and is named for Augustin Cauchy. The Cauch distribution is studied in more generality in the chapter on Special Distributions. The graph of \(f\) is known as the Witch of Agnesi , named for Maria Agnesi.
Suppose that \( X \) has the standard Cauchy distribution, and let \(M\) denote the moment generating function of \(X\). Then
- \(\E(X)\) does not exist.
- \(M(t) = \infty\) for \(t \ne 0\).
Proof
- We have seen this computation before. \( \int_a^\infty \frac{x}{\pi (1 + x^2)} \, dx = \infty \) and \( \int_{-\infty}^a \frac{x}{\pi (1 + x^2)} \, dx = -\infty \) for every \( a \in \R \), so \( \int_{-\infty}^\infty \frac{x}{\pi (1 + x^2)} \, dx \) does not exist.
- Note that \( \int_0^\infty \frac{e^{t x}}{\pi (1 + x^2) } \, dx = \infty\) if \( t \ge 0 \) and \( \int_{-\infty}^0 \frac{e^{t x}}{\pi (1 + x^2)} \, dx = \infty \) if \( t \le 0 \).
Once again, all distributions on \( \R \) have characteristic functions, and the standard Cauchy distribution has a particularly simple one.
Let \(\chi\) denote the characteristic function of \(X\). Then \(\chi(t) = e^{-\left|t\right|}\) for \(t \in \R\).
Proof
The proof of this result requires contour integrals in the complex plane, and is given in the section on the Cauchy distribution in the chapter on special distributions.
Counterexample
For the Pareto distribution, only some of the moments are finite; so course, the moment generating function cannot be finite in an interval about 0. We will now give an example of a distribution for which all of the moments are finite, yet still the moment generating function is not finite in any interval about 0. Furthermore, we will see two different distributions that have the same moments of all orders.
Suppose that Z has the standard normal distribution and let \(X = e^Z\). The distribution of \(X\) is known as the (standard) lognormal distribution . The lognormal distribution is studied in more generality in the chapter on Special Distributions. This distribution has finite moments of all orders, but infinite moment generating function.
\(X\) has probability density function \(f\) given by \[ f(x) = \frac{1}{\sqrt{2 \pi} x} \exp\left(-\frac{1}{2} \ln^2(x)\right), \quad x \gt 0 \]
- \(\E\left(X^n\right) = e^{\frac{1}{2}n^2}\) for \(n \in \N\).
- \(\E\left(e^{t X}\right) = \infty\) for \(t \gt 0\).
Proof
We use the change of variables theorem. The transformation is \( x = e^z \) so the inverse transformation is \( z = \ln x \) for \( x \in (0, \infty) \) and \( z \in \R \). Letting \( \phi \) denote the PDF of \( Z \), it follows that the PDF of \( X \) is \( f(x) = \phi(z) \, dz / dx = \phi\left(\ln x\right) \big/ x \) for \( x \gt 0 \).
- We use the moment generating function of the standard normal distribution given above : \( \E\left(X^n\right) = \E\left(e^{n Z}\right) = e^{n^2 / 2}\).
- Note that \[ \E\left(e^{t X}\right) = \E\left[\sum_{n=0}^\infty \frac{(t X)^n}{n!}\right] = \sum_{n=0}^\infty \frac{\E(X^n)}{n!} t^n = \sum_{n=0}^\infty \frac{e^{n^2 / 2}}{n!} t^n = \infty, \quad t \gt 0 \] The interchange of expected value and sum is justified since \( X \) is nonnegative. See the advanced section on properties of the integral in the chapter on Distributions for more details.
Next we construct a different distribution with the same moments as \( X \).
Let \(h\) be the function defined by \(h(x) = \sin\left(2 \pi \ln x \right)\) for \(x \gt 0\) and let \(g\) be the function defined by \(g(x) = f(x)\left[1 + h(x)\right]\) for \(x \gt 0\). Then
- \(g\) is a probability density function.
- If \( Y \) has probability density function \( g \) then \(\E\left(Y^n\right) = e^{\frac{1}{2} n^2}\) for \(n \in \N\)


2.9: Writer’s Block Just Happens to People
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- Cheryl E. Ball & Drew M. Loewe ed.
- West Virginia University via Digital Publishing Institute and West Virginia University Libraries
Author: Geoffrey V. Carter, English, Saginaw Valley State University
Whenever someone trying to write says that they are suffering from writer’s block, the first word that comes to my mind could be misunderstood as uncharitable: slacker.
At the end of this short essay, I will tell you why the word slacker pops into my mind, but first I want to share some thoughts regarding Edmund Bergler, the person who first developed the term writer’s block. Bergler was Sigmund Freud’s assistant director at the Vienna Clinic in the 1930s. Bergler made all sorts of outrageous claims, not the least of which was his ability to completely cure the malady he coined. Of all Bergler’s unsubstantiated psychological declarations, this was the one that bothered me the most. When I think about Bergler’s example, what comes to mind are writing techniques that I use to circumvent writer’s block rather than further pathologize it.
As a writer myself, I know that writing doesn’t always come easily. Fortunately, I’ve always been good at researching things before I really start writing. I was puzzled, for example, by a claim made by those who had previously studied Bergler: Bergler, evidently, had claimed that he had a 100% cure rate when it came to the malady he had invented. And yet, purportedly, Bergler never explained exactly how he treated this problem. Surely, I thought to myself, he would have mentioned a strategy somewhere. Delving deeper into such puzzling claims is an important step in a writing process and, for me, such work often alleviates the feeling of having writer’s block.
While searching through his work, I waded through Bergler’s pontifications on the frigidity of women and his rejection of homosexuality. (Bergler seriously limited his own dating pool with such proclamations.) Along the way, I found an article that gave him credit for articulating a logical connection between gambling and masochism. Perhaps Bergler wasn’t completely off his rocker, even if he wasn’t able to see past Freud’s sexual hang-ups or see fit to disclose his alleged cure for writer’s block. In all, I could only find one trivial tactic that Bergler shared—inadvertently, perhaps—in one of his clinical session anecdotes.
Before I share this anecdote, I want to mention how it reinforces some of the other strategies for overcoming writer’s block that I’ve come across. These strategies were noted both before and after Bergler’s time. The hesitation to write, after all, has been around since we’ve been trying to write. Whenever we stare at a blank page, we’re in good company.
Perhaps the funniest piece of scholarship that I’ve ever encountered about the good company of a blank page is one that was actually published in an academic journal. It can be found in the 1974 issue of the Journal of Applied Behavior Analysis , if you care to track it down yourself. (I certainly would not have delved into this journal had others not mentioned it in connection to Bergler.) When I requested this journal from the library, I most certainly wasn’t looking for a laugh. But author Dennis Upper’s entire article was summed up in the title—“The Unsuccessful Self-Treatment of a Case of Writer’s Block”—because the rest of the article (no joke) is a blank page with a footnote that it was “published without revision” following its presentation at the 81st Annual American Psychological Association Convention the year before and a note from a reviewer who vetted its inclusion into the journal: “Clearly it is the most concise manuscript I have ever seen—yet it contains sufficient detail to allow other investigators to replicate Dr. Upper’s failure.” Who knew that it would be serious business to publish a blank page in an academic journal that, at the time, had been publishing for over half a century?
The humor of this article often causes people to smile. Curiously, however, just as many people scoff at the idea of a non-article. I’m not sure why the publishing of a blank page amuses some but is an affront to others, but I have another story about writer’s block that might offer a key.
Back in the 18th century, there was a French schoolteacher and philosopher named Joseph Jacotot who put bad ideas about writer’s block to the test with the help of an illiterate Flemish locksmith. Contrary to commonly held opinion at the time, Jacotot believed that everyone—regardless of cultural hierarchy—had the capacity
for equal intelligence. His notion of intellectual emancipation led him to guide an illiterate locksmith in reading and comprehending a 17th-century French treatise. Rather than insisting his locksmith learn through a traditional a bécédaire (i.e., learning letters before learning words before learning meaning and so forth), Jacotot’s philosophy was simple: “Everything is in everything.” He operated under an assumption that it is always easier to utilize what his learners already knew. Thus the first name mentioned in the aforementioned treatise, “Calypso,” was understood by the locksmith’s recognition of the square and the round to identify what more traditional learners would label as L and O.
Jacotot’s “everything is in everything” tact is one that I’ve come to feel embodies my approach to writing: As I was exploring Bergler, I was researching a little bit of everything and incorporating a little bit of everything. I could tell Upper’s blank page was a clue, and as I listened to Jacotot’s story about the locksmith playing with names, I was getting down with a tactic only hinted at in Bergler’s work.
And this is where I come back ‘round to Bergler and an anecdote he briefly shared about one of his writer’s block patients. As it turns out, the patient said he “unlocked his [ own ] literary resources” by playing with his psychiatrist’s name: Bergler. Whether the patient was calling Bergler out as a burglar for taking his money when he was solving his own problem can’t be verified. But the key here— and yes, key is a punny reference to the illiterate locksmith—is the idea that one can facilitate writing by embracing the blank page, by remembering “everything is in everything,” and by playing with words and names. Doing so, I believe, negates the very problem of writer’s block.
These stories about writer’s block lead me to suggest that it might be useful to experiment with playing with names to get one’s writing process underway. It’s simple. By looking at your own name and the names of others, we might find puns and anagrams to help move writing along. Bergler–Burglar is simple, but as “everything is in everything,” even finding a certain glee in a name is permissible. I find such play with names and words loosens me up, and perhaps this might help others who feel frustrated or blocked too.
It all comes down to this: When faced with the process of creating something, rather than just giving up, writing about anything that comes to mind—even if it is just fooling around with words— can sometimes motivate real work. Being playful, after all, often leads to storytelling about why one is being playful. If this can be accepted, the reason why the word slacker comes to my mind is revealed as the prospect of being more than someone who simply isn’t trying hard enough.
Before I leave you with a final impression of myself as having contempt for someone struggling to create, let me clarify that my first thought, mentioned at the start of this piece, is of the movie Slacker (1991). This movie, written and directed by Richard Linklater, is one that on paper looks like a bad idea—a really bad idea. There are no big stars, no plotline, no main characters or character development, no traditional theatrical structure, no dramatic music—indeed, there’s no soundtrack at all. Slacker has no special effects or any real action, and, as such, it contains nothing that a Hollywood Feature would usually feature.
But despite the nonexistence of all these traditional elements, this film is a masterpiece of the mundane. For the entirety of the movie, for a full one hour and forty minutes, nothing of consequence happens. But it is just brilliant. Nothing happens, and yet the film works .
One character from Slacker , in particular, is an aspiring writer at a coffee shop. At his scene’s start, he is sitting, waiting for a friend. As he waits, we hear him in the background expounding on ideas for what he hopes will be the next Great American Novel. When his friend approaches his table, our aspiring writer forgoes the usual, friendly salutations and, instead, immediately enlists his friend into the fever of his writing brainstorm. The aspiring writer’s various riffs—such as calling for “a full circle aesthetic re-evaluation”—make him sound like a pseudo-intellectual blowhard who, at one point, sees himself as the next Dostoevsky. But then—after all this talk about writing—our aspiring writer reaches a moment of clarity: “Who’s ever written the great work about the immense effort required in order not to create?” This line distills the essence of Slacker into a single sentence. Linklater, of course, captures this idea in his film’s title. The line is all about turning the struggle to write back upon itself in order to create. You have to try very hard in order not to create at all.
Further Reading
For further reading on the history of writer’s block and how writing teachers have contended with this idea, see Mike Rose’s When a Writer Can’t Write (The Guilford Press), which offers a series of essays on overcoming writer’s block. Additionally, to learn more about how the play of names might be used to overcome writer’s block, see my dissertation, Rereading and Rewriting Bloc/ ks: Teaching Multi-Modal Literacies Through an Apprenticeship in Proper Names (Proquest). Jacques Rancière’s book The Ignorant Schoolmaster (Stanford University Press) and Edmund Bergler’s article “Does ‘Writer’s Block’ exist?” ( The American Imago ) offer important historical examples of blocked writing. Of course, Dennis Upper’s article “The Unsuccessful Self-Treatment of a Case of ‘Writer’s Block’” ( Journal of Applied Behavior ) needs to be seen to be believed. Better, still, is my recommendation that you link later to Richard Linklater’s movie, Slacker , and perhaps embrace the idea that telling the story about blocked writers may offer story enough.
Keywords
blank pages, play of names, puns, writer’s block
Author Bio
Geoffrey V. Carter is a tenured assistant professor of English at Saginaw Valley State University (SVSU), where he teaches undergraduate English classes and graduate courses in the communication and media administration program. His areas of interest include video culture, electracy, and post-cinematics. He has published articles in PRE/TEXT , Kairos , and Computers and Composition , and he has also edited a special issue on “Video and Participatory Cultures” for Enculturation and on “The Rhetoric of Pinball” for Itineration . He would like to thank his wife, Sara JacobsCarter, for her input on this essay and all her help with the dissertation upon which this is also based.


7.1: Introduction
Most of us know what a concert is, even if we’ve never been to one. They are common across categories of music and always follow the same basic formula: members of the public assemble at a given time and place to hear a soloist or ensemble present a prepared program of music. Concerts always tend to be presentational in nature (that is to say, there is a clear divide between performer and audience member), although behavioral norms vary across genres. Attendees at a Christian rock concert might get involved with worship, while hip-hop fans might dance, country fans might sing along, and audience members at an orchestral concert might sit in quiet contemplation or follow along with the printed score (a book of music that contains all of the orchestral parts). For the most part, concerts are intended to entertain ticket holders and to turn a profit for the artists and producers who present them.
In this chapter, we will examine four specific concerts that were staged in Europe and the United States between 1808 and 1969. We will consider the purpose for each concert, learn about the composers and producers involved in its presentation, and listen closely to a single musical work or performance. Each of these concerts was unique, and they span the gamut in terms of venue, audience, and repertoire. In order to set the stage for our encounter with concert culture, however, a brief overview of music as public entertainment is in order.
Despite the ubiquity of concerts today, musical performance as a major commercial venture has a relatively short history. In Europe, professional music first thrived in courts and churches. The wealthy staged elaborate musical presentations—such as the opera Orpheus (1607) at the court in Mantua, discussed in Chapter 4—for their own private consumption, but tickets were not put on sale. The Catholic church employed professional singers—such as Giovanni da Palestrina (1525-1594) at the Sistine chapel, discussed in Chapter 11—to provide music for worship services, but the music they produced was not intended to have entertainment value.
The first musical presentations for which members of the public could buy tickets were operas, which became available when the St. Cassiano Theater opened in Venice in 1637. Opera quickly became big business. Large crowds flocked to theaters to see the glamorous singers, fabulous costumes, and astonishing stage sets. However, there was one problem: In most places, the church authorities prohibited the performance of opera during the season of Lent. Lent, which constitutes the forty days preceding Easter, is the most solemn period in the church calendar. Members of most European Christian communities—most importantly, Catholics—were expected to abstain from frivolity and pass their time in spiritual contemplation. Opera was simply too exciting and fun.
Concerts, therefore, were first introduced as a solemn alternative to opera. The most successful early concert series was launched in Paris in 1725. It was titled the Concert Spirituel, and—as the name suggests—offered uplifting entertainment that would not offend the Catholic Church. These early concerts included a great deal of variety. In addition to choral works with a sacred message, the program was likely to include concertos, arias, and improvisations. Most of the music would have been recently composed and, despite the advertising, was not explicitly sacred. In order to replicate the thrills of opera as closely as possible, each concert began the same way as an operatic performance: The first thing on the program was always a sinfonia for orchestra in three parts, or movements, ordered fast- slow-fast. Over time, this became the symphony , perhaps the most important genre of composition for orchestra.
During the 18th and 19th centuries, concerts became more and more important in European life. At the same time, orchestral music grew in prominence, and came to be understood as being more dignified and serious than opera. The idea of a concert as commercial entertainment, however, was never confined to the orchestra. In the 1830s, for example, the Hungarian pianist Franz Liszt began to give solo piano recitals across Europe (his career and music are discussed in Chapter 9). In the last 150 years, the concert model has been increasingly adopted by traditions outside of Europe. The first Chinese orchestra, the Shanghai Municipal Orchestra, was formed in 1879, while Indian classical musicians, who had formerly been employed by courts, began to stage public concerts near the turn of the century. These developments reflected a growing reliance on capitalistic economic models throughout the world, as musicians began to rely on ticket sales rather than aristocratic patronage.
The modern concert economy works in tandem with the recording industry, which has helped performing artists to gain international fame since the early 20th century. When you choose to buy tickets to a concert, it is usually because you already know and enjoy the music you are going to hear. This is a change from the earliest concerts, which were understood as an opportunity to introduce new music to the public. Such was the case with our first example.


3.12: Reading- Characteristics of Metamorphic Rocks
A metamorphic rock used to be some other type of rock, but it was changed inside the Earth to become a new type of rock. The word metamorphism comes from ancient Greek words for “change” (meta) and “form” (morph). The type of rock that a metamorphic rock used to be, prior to metamorphism, is called the protolith. During metamorphism the mineral content and texture of the protolith are changed due to changes in the physical and chemical environment of the rock. Metamorphism can be caused by burial, tectonic stress, heating by magma, or alteration by fluids. At advanced stages of metamorphism, it is common for a metamorphic rock to develop such a different set of minerals and such a thoroughly changed texture that it is difficult to recognize what the protolith was.
A rock undergoing metamorphism remains a solid rock during the process. Rocks do not melt during most conditions of metamorphism. At the highest grade of metamorphism, rocks begin to partially melt, at which point the boundary of metamorphic conditions is surpassed and the igneous part of the rock cycle is entered.
Even though rocks remain solid during metamorphism, fluid is generally present in the microscopic spaces between the minerals. This fluid phase may play a major role in the chemical reactions that are an important part of how metamorphism occurs. The fluid usually consists largely of water.
Metamorphic rocks provide a record of the processes that occurred inside Earth as the rock was subjected to changing physical and chemical conditions. This gives the geologist literally “inside information” on what occurs within the Earth during such processes as the formation of new mountain ranges, the collision of continents, the subduction of oceanic plates, and the circulation of sea water into hot oceanic crust. Metamorphic rocks are like probes that have gone down into the Earth and come back, bringing an record of the conditions they encountered on their journey in the depths of the Earth.
In the large outcrop of metamorphic rocks in figure 1, the rocks’ platy appearance is a result of the process metamorphism. Metamorphism is the addition of heat and/or pressure to existing rocks, which causes them to change physically and/or chemically so that they become a new rock. Metamorphic rocks may change so much that they may not resemble the original rock.
Metamorphism
Any type of rock—igneous, sedimentary, or metamorphic—can become a metamorphic rock. All that is needed is enough heat and/or pressure to alter the existing rock’s physical or chemical makeup without melting the rock entirely.
Rocks change during metamorphism because the minerals need to be stable under the new temperature and pressure conditions. The need for stability may cause the structure of minerals to rearrange and form new minerals. Ions may move between minerals to create minerals of different chemical composition. Hornfels, with its alternating bands of dark and light crystals, is a good example of how minerals rearrange themselves during metamorphism. Hornfels is shown in table 1.
Extreme pressure may also lead to foliation , the flat layers that form in rocks as the rocks are squeezed by pressure (figure 2). Foliation normally forms when pressure is exerted in only one direction. Metamorphic rocks may also be non-foliated. Quartzite and limestone, shown in table 6, are nonfoliated.
The two main types of metamorphism are both related to heat within Earth:
- Regional metamorphism: Changes in enormous quantities of rock over a wide area caused by the extreme pressure from overlying rock or from compression caused by geologic processes. Deep burial exposes the rock to high temperatures.
- Contact metamorphism: Changes in a rock that is in contact with magma because of the magma’s extreme heat.
Factors that Control Metamorphism
The reason rocks undergo metamorphism is that the minerals in a rock are only stable under a limited range of pressure, temperature, and chemical conditions. When rocks are subjected to large enough changes in these factors, the minerals will undergo chemical reactions that result in their replacement by new minerals, minerals that are stable in the new conditions.
Chemical Composition of the Protolith
The type of rock undergoes metamorphism is a major factor in determing what type of metamorphic rock it becomes. In short the identify of the protolith plays a big role the identity of the metamorphic rock. A fluid phase may introduce or remove chemical substances into or out of the rock during metamorphism, but in most metamorphic rock, most of the atoms in the protolith are be present in the metamorphic rock after metamorphism; the atoms will likely be rearranged into new mineral forms within the rock. Therefore, not only does the protolith determine the initial chemistry of the metamorphic rock, most metamorphic rocks do not change their bulk (overall) chemical compositions very much during metamorphism. The fact that most metamorphic rocks retain most of their original atoms means that even if the rock was so thoroughly metamorphosed that it no longer looks at all like the protolith, the rock can be analyzed in terms of its bulk chemical composition to determine what type of rock the protolith was.
Temperature
Temperature is another major factor of metamorphism. There are two ways to think about how the temperature of a rock can be increased as a result of geologic processes.
If rocks are buried within the Earth, the deeper they go, the higher the temperatures they experience. This is because temperature inside the Earth increases along what is called the geothermal gradient, or geotherm for short. Therefore, if rocks are simply buried deep enough enough sediment, they will experience temperatures high enough to cause metamorphism. This temperature is about 200ºC (approximately 400ºF).
Tectonic processes are another way rocks can be moved deeper along the geotherm. Faulting and folding the rocks of the crust, can move rocks to much greater depth than simple burial can.
Yet another way a rock in the Earth’s crust can have its temperature greatly increased is by the intrusion of magma nearby. Magma intrusion subjects nearby rock to higher temperature with no increase in depth or pressure.
Pressure
Pressure is a measure of the stress, the physical force, being applied to the surface of a material. It is defined as the force per unit area acting on the surface, in a direction perpendicular to the surface.
Lithostatic pressure is the pressure exerted on a rock by all the surrounding rock. The source of the pressure is the weight of all the rocks above. Lithostatic pressure increases as depth within the Earth increases and is a uniform stress— the pressure applies equally in all directions on the rock.
If pressure does not apply equally in all directions, differential stress occurs. There are two types of differential stress .
Normal stress compresses (pushes together) rock in one direction, the direction of maximum stress. At the same time, in a perpendicular direction, the rock undergoes tension (stretching), in the direction of minimum stress.
Shear stress pushes one side of the rock in a direction parallel to the side, while at the same time, the other side of the rock is being pushed in the opposite direction.
Differential stress has a major influence on the the appearance of a metamorphic rock. Differential stress can flatten pre-existing grains in the rock, as shown in the diagram below.
Metamorphic minerals that grow under differential stress will have a preferred orientation if the minerals have atomic structures that tend to make them form either flat or elongate crystals. This will be especially apparent for micas or other sheet silicates that grow during metamorphism, such as biotite, muscovite, chlorite, talc, or serpentine. If any of these flat minerals are growing under normal stress, they will grow with their sheets oriented perpendicular to the direction of maximum compression. This results in a rock that can be easily broken along the parallel mineral sheets. Such a rock is said to be foliated, or to have foliation.
Fluids
Any open space between the mineral grains in a rock, however microscopic, may contain a fluid phase. Most commonly, if there is a fluid phase in a rock during metamorphism, it will be a hydrous fluid, consisting of water and things dissolved in the water. Less commonly, it may be a carbon dioxide fluid or some other fluid. The presence of a fluid phase is a major factor during metamorphism because it helps determine which metamorphic reactions will occur and how fast they will occur. The fluid phase can also influence the rate at which mineral crystals deform or change shape. Most of this influence is due to the dissolved ions that pass in and out of the fluid phase. If during metamorphism enough ions are introduced to or removed from the rock via the fluid to change the bulk chemical composition of the rock, the rock is said to have undergone metasomatism. However, most metamorphic rocks do not undergo sufficient change in their bulk chemistry to be considered metasomatic rocks.
Time
Most metamorphism of rocks takes place slowly inside the Earth. Regional metamorphism takes place on a timescale of millions of years. Metamorphism usually involves slow changes to rocks in the solid state, as atoms or ions diffuse out of unstable minerals that are breaking down in the given pressure and temperature conditions and migrate into new minerals that are stable in those conditions. This type of chemical reaction takes a long time.
Grades of Metamorphism
Metamorphic grade refers to the general temperature and pressure conditions that prevailed during metamorphism. As the pressure and temperature increase, rocks undergo metamorphism at higher metamorphic grade. Rocks changing from one type of metamorphic rock to another as they encounter higher grades of metamorphism are said to be undergoing prograde metamorphism.
Low-grade metamorphism takes place at approximately 200–320 ºC and relatively low pressure. This is not far beyond the conditions in which sediments get lithified into sedimentary rocks, and it is common for a low-grade metamorphic rock to look somewhat like its protolith. Low grade metamorphic rocks tend to characterized by an abundance of hydrous minerals, minerals that contain water within their crystal structure. Examples of low grade hydrous minerals include clay, serpentine, and chlorite. Under low grade metamorphism many of the metamorphic minerals will not grow large enough to be seen without a microscope.
Medium-grade metamorphism takes place at approximately at 320–450 ºC and at moderate pressures. Low grade hydrous minerals are replaced by micas such as biotite and muscovite, and non-hydrous minerals such as garnet may grow. Garnet is an example of a mineral which may form porphyroblasts, metamorphic mineral grains that are larger in size and more equant in shape (about the same diameter in all directions), thus standing out among the smaller, flatter, or more elongate minerals.
High-grade metamorphism takes place at temperatures above about 450 ºC. Micas tend to break down. New minerals such as hornblende will form, which is stable at higher temperatures. However, as metamorphic grade increases to even higher grade, all hydrous minerals, which includes hornblende, may break down and be replaced by other, higher-temperature, non-hydrous minerals such as pyroxene.
Index Minerals
Index minerals, which are indicators of metamorphic grade. In a given rock type, which starts with a particular chemical composition, lower-grade index minerals are replaced by higher-grade index minerals in a sequence of chemical reactions that proceeds as the rock undergoes prograde metamorphism. For example, in rocks made of metamorphosed shale, metamorphism may prograde through the following index minerals:
- chlorite characterizes the lowest regional metamorphic grade
- biotite replaces chlorite at the next metamorphic grade, which could be considered medium-low grade
- garnet appears at the next metamorphic grade, medium grade
- staurolite marks the next metamorphic grade, which is medium-high grade
- sillimanite is a characteristic mineral of high grade metamorphic rocks
Index minerals are used by geologists to map metamorphic grade in regions of metamorphic rock. A geologist maps and collects rock samples across the region and marks the geologic map with the location of each rock sample and the type of index mineral it contains. By drawing lines around the areas where each type of index mineral occurs, the geologist delineates the zones of different metamorphic grades in the region. The lines are known as isograds.
Types of Metamorphism
Regional Metamorphism
Regional metamorphism occurs where large areas of rock are subjected to large amounts of differential stress for long intervals of time, conditions typically associated with mountain building. Mountain building occurs at subduction zones and at continental collision zones where two plates each bearing continental crust, converge upon each other.
Most foliated metamorphic rocks—slate, phyllite, schist, and gneiss—are formed during regional metamorphism. As the rocks become heated at depth in the Earth during regional metamorphism they become ductile, which means they are relatively soft even though they are still solid. The folding and deformation of the rock while it is ductile may greatly distort the original shapes and orientations of the rock, producing folded layers and mineral veins that have highly deformed or even convoluted shapes. The diagram below shows folds forming during an early stage of regional metamorphism, along with development of foliation, in response to normal stress.
The photograph below shows high-grade metamorphic rock that has undergone several stages of foliation development and folding during regional metamorphism, and may even have reached such a high temperature that it began to melt.
Contact Metamorphism
Contact metamorphism occurs to solid rock next to an igneous intrusion and is caused by the heat from the nearby body of magma. Because contact metamorphism is not caused by changes in pressure or by differential stress, contact metamorphic rocks do not become foliated. Where intrusions of magma occur at shallow levels of the crust, the zone of contact metamorphism around the intrusion is relatively narrow, sometimes only a few m (a few feet) thick, ranging up to contact metamorphic zones over 1000 m (over 3000 feet) across around larger intrusions that released more heat into the adjacent crust. The zone of contact metamorphism surrounding an igneous intrusion is called the metamorphic aureole. The rocks closest to the contact with the intrusion are heated to the highest temperatures, so the metamorphic grade is highest there and diminishes with increasing distance away from the contact. Because contact metamorphism occurs at shallow to moderate depths in the crust and subjects the rocks to temperatures up to the verge of igneous conditions, it is sometimes referred to as high-temperature, low-pressure metamorphism. Hornfels, which is a hard metamorphic rock formed from fine-grained clastic sedimentary rocks, is a common product of contact metamorphism.
Hydrothermal Metamorphism
Hydrothermal metamorphism is the result of extensive interaction of rock with high-temperature fluids. The difference in composition between the existing rock and the invading fluid drives the chemical reactions. The hydrothermal fluid may originate from a magma that intruded nearby and caused fluid to circulate in the nearby crust, from circulating hot groundwater, or from ocean water. If the fluid introduces substantal amounts of ions into the rock and removes substantial amounts of ions from it, the fluid has metasomatized the rock—changed its chemical composition.
Ocean water that penetrates hot, cracked oceanic crust and circulates as hydrothermal fluid in ocean floor basalts produces extensive hydrothermal metamorphism adjacent to mid-ocean spreading ridges and other ocean-floor volcanic zones. Much of the basalt subjected to this type of metamorphism turns into a type of metamorphic rock known as greenschist. Greenschist contains a set of minerals, some of them green, which may include chlorite, epidote, talc, Na-plagioclase, or actinolite. The fluids eventually escape through vents in the ocean floor known as black smokers, producing thick deposits of minerals on the ocean floor around the vents.
Burial Metamorphism
Burial metamorphism occurs to rocks buried beneath sediments to depths that exceed the conditions in which sedimentary rocks form. Because rocks undergoing burial metamorphism encounter the uniform stress of lithostatic pressure, not differential pressure, they do not develop foliation. Burial metamorphism is the lowest grade of metamorphism. The main type of mineral that usually grows during burial metamorphism is zeolite, a group of low-density silicate minerals. It usually requires a strong microscope see the small grains of zeolite minerals that form during burial metamorphism.
Subduction Zone Metamorphism
During subduction, a tectonic plate, consisting of oceanic crust and lithospheric mantle, is recycled back into the deeper mantle. In most subduction zones the subducting plate is relatively cold compared with the high temperature it had when first formed at a mid-ocean spreading ridge. Subduction takes the rocks to great depth in the Earth relatively quickly. This produces a characteristic type of metamorphism, sometimes called high-pressure, low-temperature (high-P, low-T) metamorphism, which only occurs deep in a subduction zone. In oceanic basalts that are part of a subducting plate, the high-P, low-T conditions create a distinctive set of metamorphic minerals including a type of amphibole, called glaucophane, that has a blue color. Blueschist is the name given to this type of metamorphic rock. Blueschist is generally interpreted as having been produced within a subduction zone, even if the plate boundaries have subsequently shifted and that location is no longer at a subduction zone.
Metamorphic Facies
Much as the minerals and textures of sedimentary rocks can be used as windows to see into the environment in which the sediments were deposited on the Earth’s surface, the minerals and textures of metamorphic rocks provide windows through which we view the conditions of pressure, temperature, fluids, and stress that occurred inside the Earth during metamorphism. The pressure and temperature conditions under which specific types of metamorphic rocks form has been determined by a combination labratory experiments, physics-based theoretical calculations, along with evidence in the textures of the rocks and their field relations as recorded on geologic maps. The knowledge of temperatures and pressures at which particular types of metamorphic rocks form led to the concept of metamorphic facies. Each metamorphic facies is represented by a specific type of metamorphic rock that forms under a specific pressure and temperature conditions.
Even though the name of the each metamorphic facies is taken from a type of rock that forms under those conditions, that is not the only type of rock that will form in those conditions. For example, if the protolith is basalt, it will turn into greenschist under greenschist facies conditions, and that is what facies is named for. However, if the protolith is shale, a muscovite-biotite schist, which is not green, will form instead. If it can be determined that a muscovite-biotite schist formed at around 350ºC temperature and 400 MPa pressure, it can be stated that the rock formed in the greenschist facies, even though the rock is not itself a greenschist.
The diagram below shows metamorphic facies in terms of pressure and temperature condiditons inside the Earth. Earth’s surface conditions are near the top left corner of the graph at about 15ºC which is the average temperature at Earth’s surface and 0.1 MPa (megapascals), which is about the average atmospheric pressure on the Earth’s surface. Just as atmospheric pressure comes from the weight of all the air above a point on the Earth’s surface, pressure inside the Earth comes from the weight of all the rock above a given depth. Rocks are much denser than air and MPa is the unit most commonly uses to express pressures inside the Earth. One MPa equals nearly 10 atmospheres. A pressure of 1000 MPa corresponds to a depth of about 35 km inside the Earth. Although pressure inside the Earth is determined by the depth, temperature depends on more than depth. Temperature depends on the heat flow, which varies from location to location. The way temperature changes with depth inside the Earth is called the geothermal gradient, geotherm for short. In the diagram below, three different geotherms are marked with dashed lines. The three geotherms represent different geological settings in the Earth.
High-pressure, low-temperature geotherms occurs in subduction zones. As the diagram shows, rocks undergoing prograde metamorphism in subduction zones will be subjected to zeolite, blueschist, and ultimately eclogite facies conditions.
High-temperature, low-pressure geotherms occur in the vicinity of igneous intrusions in the shallow crust, underlying a volcanically active area. Rocks that have their pressure and temperature conditions increased along such a geotherm will metamorphose in the hornfels facies and, if it gets hot enough, in the granulite facies.
Blueschist facies and hornfels facies are associated with unusual geothermal gradients. The most common conditions in the Earth are found along geotherms between those two extremes. Most regional metamorphic rocks are formed in conditions within this range of geothermal gradients, passing through the greenschist facies to the amphibolites facies. At the maximum pressures and temperatures the rocks may encounter within the Earth in this range of geotherms, they will enter either the granulite or eclogite facies. Regionally metamorphosed rocks that contain hydrous fluids will begin to melt before they pass beyond the amphibolite facies.
Types of Metamorphic Rocks
Metamorphic rock fall into two categories, foliated and unfoliated. Most foliated metamorphic rocks originate from regional metamorphism. Some unfoliated metamorphic rocks, such as hornfels, originate only by contact metamorphism, but others can originate either by contact metamorphism or by regional metamorphism. Quartz and marble are prime examples of unfoliated that can be produced by either regional or contact metamorphism. Both rock types consist of metamorphic minerals that do not have flat or elongate shapes and thus cannot become layered even if they are produced under differential stress.
A geologist working with metamorphic rocks collects the rocks in the field and looks for the patterns the rocks form in outcrops as well as how those outcrops are related to other types of rock with which they are in contact. Field evidence is often required to know for sure whether rocks are products of regional metamorphism, contact metamorphism, or some other type of metamorphism. If only looking at rock samples in a laboratory, one can be sure of the type of metamorphism that produced a foliated metamorphic rock such as schist or gneiss, or a hornfels, which is unfoliated, but one cannot be sure of the type of metamorphism that produced an unfoliated marble or quartzite.
Foliated Metamorphic Rocks
Foliated metamorphic rocks are named for their style of foliation. However, a more complete name of each particular type of foliated metamorphic rock includes the main minerals that the rock comprises, such as biotite-garnet schist rather than just schist.
- slate—slates form at low metamorphic grade by the growth of fine-grained chlorite and clay minerals. The preferred orientation of these sheet silicates causes the rock to easily break along parallel planes, giving the rock a slaty cleavage. Some slate breaks into such extensively flat sheets of rock that it is used as the base of pool tables, beneath a layer of rubber and felt. Roof tiles are also sometimes made of slate.
- phyllite—phyllite is a low-medium grade regional metamorphic rock in which the clay minerals and chlorite have been at least partly replaced by mica mica minerals, muscovite and biotite. This gives the surfaces of phyllite a satiny luster, much brighter than the surface of a piece of slate. It is also common for the differential stresses under which phyllite forms to have produced a set of folds in the rock, making the foliation surfaces wavy or irregular, in contrast to the often perfectly flat surfaces of slaty cleavage.
- schist—the size of mineral crystals tends to grow larger with increasing metamorphic grade. Schist is a product of medium grades of metamorphism and is characterized by visibly prominent, parallel sheets of mica or similar sheet silicates, usually either muscovite or biotite, or both. In schist, the sheets of mica are usually arranged in irregular planes rather than perfectly flat planes, giving the rock a schistose foliation (or simply schistosity). Schist often contains more than just micas among its minerals, such as quartz, feldspars, and garnet.
- amphibolite—a poorly foliated to unfoliated mafic metamorphic rock, usually consisting largely of the common black amphibole known as hornblende, plus plagioclase, plus or minus biotite and possibly other minerals; it usually does not contain any quartz. Amphibolite forms at medium-high metamorphic grades. Amphibolite is also listed below in the section on unfoliated metamorphic rocks.
- gneiss—like the word schist, the word gneiss is originated from the German language; it is pronounced “nice.” As metamorphic grade continue to increase, sheet silicates become unstable and dark minerals such as hornblende or pyroxene start to grow. The dark-colored minerals tend to form separate bands or stripes in the rock, giving it a gneissic foliation of dark and light streaks. Gneiss is a high-grade metamorphic rock. Many types of gneiss look somewhat like granite, except that the gneiss has dark and light stripes whereas in granite randomly oriented and distributed minerals with no stripes or layers.
- migmatite—a combination of high-grade regional metamorphic rock – usually gneiss or schist – and granitic igneous rock. The granitic rock in migmatite probably originated from partial melting of some of the metamorphic rock, though in some migmatites the granite may have intruded the rock from deeper in the crust. In migmatite you can see metamorphic rock that has reached the limits of metamorphism and begun transitioning into the igneous stage of the rock cycle by melting to form magma.
Names of different styles of foliation come from the common rocks that exhibit such foliation:
- slate has slaty foliation
- phyllite has phyllitic foliation
- schist has schistose foliation
- gneiss has gneissic foliation (also called gneissose foliation)
Nonfoliated Metamorphic Rocks
Nonfoliated metamorphic rocks lack a planar (oriented) fabric, either because the minerals did not grow under differential stress, or because the minerals that grew during metamorphism are not minerals that have elongate or flat shapes. Because they lack foliation, these rocks are named entirely on the basis of their mineralogy.
- hornfels—hornfels are very hard rocks formed by contact metamorphism of shale, siltstone, or sandstone. The heat from the nearby magma “bakes” the sedimentary rocks and recrystallizes the minerals in them into a new texture that no longer breaks easily along the original sedimentary bedding planes. Depending on the composition of the rock and the temperature reached, minerals indicative of high metamorphic grade such as pyroxene may occur in some hornfels, though many hornfels have minerals indicating medium grade metamorphism.
- amphibolite—amphibolites are dark-colored rocks with amphibole, usually the common black amphibole known as hornblende, as their most abundant mineral, along with plagioclase and possibly other minerals, though usually no quartz. Amphibolites are poorly foliated to unfoliated and form at medium to medium-high grades of metamorphism from basalt or gabbro.
- quartzite—quartzite is a metamorphic rock made almost entirely of quartz, for which the protolith was quartz arenite. Because quartz is stable over a wide range of pressure and temperature, little or no new minerals form in quartzite during metamorphism. Instead, the quartz grains recrystallize into a denser, harder rock than the original sandstone. If struck by a rock hammer, quartzite will commonly break right through the quartz grains, rather than around them as when quartz arenite is broken.
- marble—marble is a metamorphic rock made up almost entirely of either calcite or dolomite, for which the protolith was either limestone or dolostone, respectively. Marbles may have bands of different colors which were deformed into convoluted folds while the rock was ductile. Such marble is often used as decorative stone in buildings. Some marble, which is considered better quality stone for carving into statues, lacks color bands.
| Table 1. Common Metamorphic Rocks and Their Parent Rock | |||
|---|---|---|---|
| Picture | Rock Name | Type of Metamorphic Rock | Comments |
| Slate | Foliated | Metamorphism of shale | |
| Phyllite | Foliated | Metamorphism of slate, but under greater heat and pressure than slate | |
| Schist | Foliated | Often derived from metamorphism of claystone or shale; metamorphosed under more heat and pressure than phyllite | |
| Gneiss | Foliated | Metamorphism of various different rocks, under extreme conditions of heat and pressure | |
| Hornfels | Non-foliated | Contact metamorphism of various different rock types | |
| Quartzite | Non-foliated | Metamorphism of sandstone | |
| Marble | Non-foliated | Metamorphism of limestone | |
| Metaconglomerate | Non-foliated | Metamorphism of conglomerate |
Metamorphic Rock Classification
| Foliated Metamorphic Rocks | ||||
|---|---|---|---|---|
| Crystal Size | Mineralogy | Protolith | Metamorphism | Rock Name |
| very fine | clay minerals | shale | low grade regional | slate |
| fine | clay minerals, biotite, muscovite | shale | low grade regional | phyllite |
| medium to coarse | biotite, muscovite, quartz, garnet, plagioclase | shale, basalt | medium grade regional | schist |
| medium to coarse | amphibole, plagioclase, biotite | basalt | medium grade regional |
amphibolite
(Note: may be unfoliated) |
| medium to coarse | plagioclase, orthoclase, quartz, biotite, amphibole, pyroxene | basalt, granite, shale | high grade regional | gneiss |
| Unfoliated Metamorphic Rocks | ||||
| Crystal Size | Mineralogy | Protolith | Metamorphism | Rock Name |
| fine to coarse | quartz | sandstone | regional or contact | quartzite |
| fine to coarse | calcite | limestone | regional or contact | marble |
| fine | pyroxene, amphibole, plagioclase | shale | contact | hornfels |
Note that not all minerals listed in the mineralogy column will be present in every rock of that type and that some rocks may have minerals not listed here.
Uses of Metamorphic Rocks
Quartzite and marble are commonly used for building materials and artwork. Marble is beautiful for statues and decorative items such as vases (see an example in figure 3). Ground up marble is also a component of toothpaste, plastics, and paper.
Quartzite is very hard and is often crushed and used in building railroad tracks (see figure 4). Schist and slate are sometimes used as building and landscape materials. Graphite, the “lead” in pencils, is a mineral commonly found in metamorphic rocks.
Contributors and Attributions
- Revision and Adaptation. Authored by : Kimberly Schulte and Lumen Learning. Provided by : Lumen Learning. License : CC BY: Attribution
- Basics -- Metamorphic Rocks. Authored by : Ralph L. Dawes and Cheryl D. Dawes. Provided by : Wenatchee Valley College. Located at : http://commons.wvc.edu/rdawes/G101OCL/Basics/metamorphic.html . Project : Geology 101 - Introduction to Physical Geology. License : CC BY: Attribution
- HS Rocks. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-High-School/section/4.0/ . License : CC BY-NC: Attribution-NonCommercial