4.7: Conditional Expected Value
As usual, our starting point is a random experiment modeled by a probability space \((\Omega, \mathscr F, \P)\). So to review, \( \Omega \) is the set of outcomes, \( \mathscr F \) the collection of events, and \( \P \) the probability measure on the sample space \((\Omega, \mathscr F)\). Suppose next that \(X\) is a random variable taking values in a set \(S\) and that \(Y\) is a random variable taking values in \(T \subseteq \R\). We assume that either \(Y\) has a discrete distribution, so that \(T\) is countable, or that \(Y\) has a continuous distribution so that \(T\) is an interval (or perhaps a union of intervals). In this section, we will study the conditional expected value of \(Y\) given \(X\), a concept of fundamental importance in probability. As we will see, the expected value of \(Y\) given \(X\) is the function of \(X\) that best approximates \(Y\) in the mean square sense. Note that \(X\) is a general random variable, not necessarily real-valued, but as usual, we will assume that either \(X\) has a discrete distribution, so that \(S\) is countable or that \(X\) has a continuous distribution on \(S \subseteq \R^n\) for some \(n \in \N_+\). In the latter case, \(S\) is typically a region defined by inequalites involving elementary functions. We will also assume that all expected values that are mentioned exist (as real numbers).
Basic Theory
Definitions
Note that we can think of \((X, Y)\) as a random variable that takes values in the Cartesian product set \(S \times T\). We need recall some basic facts from our work with joint distributions and conditional distributions.
We assume that \( (X, Y) \) has joint probability density function \( f \) and we let \(g\) denote the (marginal) probability density function \( X \). Recall that if \(Y\) has a discrte distribution then \[ g(x) = \sum_{y \in T} f(x, y), \quad x \in S \] and if \(Y\) has a continuous distribution then \[ g(x) = \int_T f(x, y) \, dy, \quad x \in S \] In either case, for \( x \in S \), the conditional probability density function of \( Y \) given \( X = x \) is defined by \[ h(y \mid x) = \frac{f(x, y)}{g(x)}, \quad y \in T \]
We are now ready for the basic definitions:
For \( x \in S \), the conditional expected value of \(Y\) given \(X = x \in S\) is simply the mean computed relative to the conditional distribution. So if \(Y\) has a discrete distribution then \[E(Y \mid X = x) = \sum_{y \in T} y h(y \mid x), \quad x \in S\] and if \(Y\) has a continuous distribution then \[ \E(Y \mid X = x) = \int_T y h(y \mid x) \, dy, \quad x \in S \]
- The function \(v: S \to \R\) defined by \( v(x) = \E(Y \mid X = x)\) for \( x \in S \) is the regression function of \(Y\) based on \(X\).
- The random variable \(v(X)\) is called the conditional expected value of \(Y\) given \(X\) and is denoted \(\E(Y \mid X)\).
Intuitively, we treat \(X\) as known, and therefore not random, and we then average \(Y\) with respect to the probability distribution that remains. The advanced section on conditional expected value gives a much more general definition that unifies the definitions given here for the various distribution types.
Properties
The most important property of the random variable \(\E(Y \mid X)\) is given in the following theorem. In a sense, this result states that \( \E(Y \mid X) \) behaves just like \( Y \) in terms of other functions of \( X \), and is essentially the only function of \( X \) with this property.
The fundamental property
- \( \E\left[r(X) \E(Y \mid X)\right] = \E\left[r(X) Y\right] \) for every function \( r: S \to \R \).
- If \(u: S \to \R\) satisfies \(\E[r(X) u(X)] = \E[r(X) Y]\) for every \(r: S \to \R\) then \( \P\left[u(X) = \E(Y \mid X)\right] = 1 \).
Proof
We give the proof in the continuous case. The discrete case is analogous, with sums replacing integrals.
- From the change of variables theorem for expected value, \begin{align} \E\left[r(X) \E(Y \mid X)\right] & = \int_S r(x) \E(Y \mid X = x) g(x) \, dx = \int_S r(x) \left(\int_T y h(y \mid x) \, dy \right) g(x) \, dx\\ & = \int_S \int_T r(x) y h(y \mid x) g(x) \, dy \, dx = \int_{S \times T} r(x) y f(x, y) \, d(x, y) = \E[r(X) Y] \end{align}
- Suppose that \( u_1: S \to \R \) and \( u_2: S \to \R \) satisfy the condition in (b). Define \(r: S \to \R\) by \(r(x) = \bs 1[u_1(x) \gt u_2(x)]\). Then by assumption, \(\E\left[r(X) u_1(X)\right] = \E\left[r(X) Y\right] = \E\left[r(X) u_2(X)\right]\) But if \( \P\left[u_1(X) \gt u_2(X)\right] \gt 0 \) then \( \E\left[r(X) u_1(X)\right] \gt \E\left[r(X) u_2(X)\right] \), a contradiction. Hence we must have \( \P\left[u_1(X) \gt u_2(X)\right] = 0 \) and by a symmetric argument, \( \P[u_1(X) \lt u_2(X)] = 0 \).
Two random variables that are equal with probability 1 are said to be equivalent. We often think of equivalent random variables as being essentially the same object, so the fundamental property above essentially characterizes \( \E(Y \mid X) \). That is, we can think of \( \E(Y \mid X) \) as any random variable that is a function of \( X \) and satisfies this property. Moreover the fundamental property can be used as a definition of conditional expected value, regardless of the type of the distribution of \((X, Y)\). If you are interested, read the more advanced treatment of conditional expected value.
Suppose that \( X \) is also real-valued. Recall that the best linear predictor of \( Y \) based on \( X \) was characterized by property (a), but with just two functions: \( r(x) = 1 \) and \( r(x) = x \). Thus the characterization in the fundamental property is certainly reasonable, since (as we show below) \( \E(Y \mid X) \) is the best predictor of \( Y \) among all functions of \( X \), not just linear functions.
The basic property is also very useful for establishing other properties of conditional expected value. Our first consequence is the fact that \( Y \) and \( \E(Y \mid X) \) have the same mean.
\(\E\left[\E(Y \mid X)\right] = \E(Y)\).
Proof
Let \(r\) be the constant function 1 in the basic property .
Aside from the theoretical interest, this theorem is often a good way to compute \(\E(Y)\) when we know the conditional distribution of \(Y\) given \(X\). We say that we are computing the expected value of \(Y\) by conditioning on \(X\).
For many basic properties of ordinary expected value, there are analogous results for conditional expected value. We start with two of the most important: every type of expected value must satisfy two critical properties: linearity and monotonicity. In the following two theorems, the random variables \( Y \) and \( Z \) are real-valued, and as before, \( X \) is a general random variable.
Linear Properties
- \(\E(Y + Z \mid X) = \E(Y \mid X) + \E(Z \mid X)\).
- \(\E(c \, Y \mid X) = c \, \E(Y \mid X)\)
Proof
- Note that \( \E(Y \mid X) + \E(Z \mid X) \) is a function of \( X \). If \( r: S \to \R \) then \[ \E\left(r(x) \left[\E(Y \mid X) + \E(Z \mid X)\right]\right) = \E\left[r(X) \E(Y \mid X)\right] + \E\left[r(X) \E(Z \mid X)\right] = E\left[r(X) Y\right] + \E\left[r(X) Z\right] = \E\left[r(X) (Y + Z)\right] \] Hence the result follows from the basic property .
- Note that \( c \E(Y \mid X) \) is a function of \( X \). If \( r: S \to \R \) then \[ \E\left[r(X) c \E(Y \mid X)\right] = c \E\left[r(X) \E(Y \mid X)\right] = c \E\left[r(X) Y\right] = \E\left[r(X) (c Y)\right] \] Hence the result follows from the basic property
Part (a) is the additive property and part (b) is the scaling property. The scaling property will be significantly generalized below in (8) .
Positive and Increasing Properties
- If \(Y \ge 0\) then \(\E(Y \mid X) \ge 0\).
- If \(Y \le Z\) then \(\E(Y \mid X) \le \E(Z \mid X)\).
- \( \left|\E(Y \mid X)\right| \le \E\left(\left|Y\right| \mid X\right)\)
Proof
- This follows directly from the definition.
- Note that if \( Y \le Z \) then \( Y - Z \ge 0 \) so by (a) and linearity, \[ \E(Y - Z \mid X) = \E(Y \mid X) - \E(Z \mid X) \ge 0 \]
- Note that \( -\left|Y\right| \le Y \le \left|Y\right| \) and hence by (b) and linearity, \(-\E\left(\left|Y\right| \mid X \right) \le \E(Y \mid X) \le \E\left(\left|Y\right| \mid X\right)\).
Our next few properties relate to the idea that \( \E(Y \mid X) \) is the expected value of \( Y \) given \( X \). The first property is essentially a restatement of the fundamental property .
If \(r: S \to \R\), then \(Y - \E(Y \mid X)\) and \(r(X)\) are uncorrelated.
Proof
Note that \( Y - \E(Y \mid X) \) has mean 0 by the mean property . Hence, by the basic property , \[ \cov\left[Y - \E(Y \mid X), r(X)\right] = \E\left\{\left[Y - \E(Y \mid X)\right] r(X)\right\} = \E\left[Y r(X)\right] - \E\left[\E(Y \mid X) r(X)\right] = 0 \]
The next result states that any (deterministic) function of \(X\) acts like a constant in terms of the conditional expected value with respect to \(X\).
If \(s: S \to \R\) then \[ \E\left[s(X)\,Y \mid X\right] = s(X)\,\E(Y \mid X) \]
Proof
Note that \( s(X) \E(Y \mid X) \) is a function of \( X \). If \( r: S \to \R \) then \[ \E\left[r(X) s(X) \E(Y \mid X)\right] = \E\left[r(X) s(X) Y\right] \] So the result now follow from the basic property .
The following rule generalizes theorem (8) and is sometimes referred to as the substitution rule for conditional expected value.
If \(s: S \times T \to \R\) then \[ \E\left[s(X, Y) \mid X = x\right] = \E\left[s(x, Y) \mid X = x\right] \]
In particular, it follows from (8) that \(\E[s(X) \mid X] = s(X)\). At the opposite extreme, we have the next result: If \(X\) and \(Y\) are independent, then knowledge of \(X\) gives no information about \(Y\) and so the conditional expected value with respect to \(X\) reduces to the ordinary (unconditional) expected value of \(Y\).
If \(X\) and \(Y\) are independent then \[ \E(Y \mid X) = \E(Y) \]
Proof
Trivially, \( \E(Y) \) is a (constant) function of \( X \). If \( r: S \to \R \) then \( \E\left[\E(Y) r(X)\right] = \E(Y) \E\left[r(X)\right] = \E\left[Y r(X)\right] \), the last equality by independence. Hence the result follows from the basic property .
Suppose now that \(Z\) is real-valued and that \(X\) and \(Y\) are random variables (all defined on the same probability space, of course). The following theorem gives a consistency condition of sorts. Iterated conditional expected values reduce to a single conditional expected value with respect to the minimum amount of information. For simplicity, we write \( \E(Z \mid X, Y) \) rather than \( \E\left[Z \mid (X, Y)\right] \).
Consistency
- \(\E\left[\E(Z \mid X, Y) \mid X\right] = \E(Z \mid X)\)
- \(\E\left[\E(Z \mid X) \mid X, Y\right] = \E(Z \mid X)\)
Proof
- Suppose that \( X \) takes values in \( S \) and \( Y \) takes values in \( T \), so that \( (X, Y) \) takes values in \( S \times T \). By definition, \( \E(Z \mid X) \) is a function of \( X \). If \( r: S \to \R \) then trivially \( r \) can be thought of as a function on \( S \times T \) as well. Hence \[ \E\left[r(X) \E(Z \mid X)\right] = \E\left[r(X) Z\right] = \E\left[r(X) \E(Z \mid X, Y)\right] \] It follows from the basic property that \(\E\left[\E(Z \mid X, Y) \mid X\right] = \E(Z \mid X) \).
- Note that since \( \E(Z \mid X) \) is a function of \( X \), it is trivially a function of \( (X, Y) \). Hence from (8) , \( \E\left[\E(Z \mid X) \mid X, Y\right] = \E(Z \mid X) \).
Finally we show that \( \E(Y \mid X) \) has the same covariance with \( X \) as does \( Y \), not surprising since again, \( \E(Y \mid X) \) behaves just like \( Y \) in its relations with \( X \).
\(\cov\left[X, \E(Y \mid X)\right] = \cov(X, Y)\).
Proof
\( \cov\left[X, \E(Y \mid X)\right] = \E\left[X \E(Y \mid X)\right] - \E(X) \E\left[\E(Y \mid X)\right] \). But \( \E\left[X \E(Y \mid X)\right] = \E(X Y) \) by basic property , and \( \E\left[\E(Y \mid X)\right] = \E(Y) \) by the mean property . Hence \( \cov\left[X, \E(Y \mid X)\right] = \E(X Y) - \E(X) \E(Y) = \cov(X, Y) \).
Conditional Probability
The conditional probability of an event \(A\), given random variable \(X\) (as above), can be defined as a special case of the conditional expected value. As usual, let \(\bs 1_A\) denote the indicator random variable of \(A\).
If \(A\) is an event, defined \[ \P(A \mid X) = \E\left(\bs{1}_A \mid X\right) \]
Here is the fundamental property for conditional probability:
The fundamental property
- \( \E\left[r(X) \P(A \mid X)\right] = \E\left[r(X) \bs{1}_A\right] \) for every function \( r: S \to \R \).
- If \( u: S \to \R \) and \( u(X) \) satisfies \( \E[r(X) u(X)] = \E\left[r(X) \bs 1_A\right] \) for every function \( r: S \to \R \), then \( \P\left[u(X) = \P(A \mid X)\right] = 1 \).
For example, suppose that \( X \) has a discrete distribution on a countable set \( S \) with probability density function \( g \). Then (a) becomes \[ \sum_{x \in S} r(x) \P(A \mid X = x) g(x) = \sum_{x \in S} r(x) \P(A, X = x) \] But this is obvious since \( \P(A \mid X = x) = \P(A, X = x) \big/ \P(X = x) \) and \( g(x) = \P(X = x) \). Similarly, if \( X \) has a continuous distribution on \( S \subseteq \R^n \) then (a) states that \[ \E\left[r(X) \bs{1}_A\right] = \int_S r(x) \P(A \mid X = x) g(x) \, dx \]
The properties above for conditional expected value, of course, have special cases for conditional probability.
\(\P(A) = \E\left[\P(A \mid X)\right]\).
Proof
This is a direct result of the mean property , since \( \E(\bs{1}_A) = \P(A) \).
Again, the result in the previous exercise is often a good way to compute \(\P(A)\) when we know the conditional probability of \(A\) given \(X\). We say that we are computing the probability of \(A\) by conditioning on \(X\). This is a very compact and elegant version of the conditioning result given first in the section on Conditional Probability in the chapter on Probability Spaces and later in the section on Discrete Distributions in the Chapter on Distributions.
The following result gives the conditional version of the axioms of probability.
Axioms of probability
- \( \P(A \mid X) \ge 0 \) for every event \( A \).
- \( \P(\Omega \mid X) = 1 \)
- If \( \{A_i: i \in I\} \) is a countable collection of disjoint events then \( \P\left(\bigcup_{i \in I} A_i \bigm| X\right) = \sum_{i \in I} \P(A_i \mid X)\).
Details
There are some technical issues involving the countable additivity property (c). The conditional probabilities are random variables, and so for a given collection \(\{A_i: i \in I\}\), the left and right sides are the same with probability 1. We will return to this point in the more advanced section on conditional expected value
From the last result, it follows that other standard probability rules hold for conditional probability given \( X \). These results include
- the complement rule
- the increasing property
- Boole's inequality
- Bonferroni's inequality
- the inclusion-exclusion laws
The Best Predictor
The next result shows that, of all functions of \(X\), \(\E(Y \mid X)\) is closest to \(Y\), in the sense of mean square error. This is fundamentally important in statistical problems where the predictor vector \(X\) can be observed but not the response variable \(Y\). In this subsection and the next, we assume that the real-valued random variables have finite variance.
If \(u: S \to \R\), then
- \(\E\left(\left[\E(Y \mid X) - Y\right]^2\right) \le \E\left(\left[u(X) - Y\right]^2\right)\)
- Equality holds in (a) if and only if \(u(X) = \E(Y \mid X)\) with probability 1.
Proof
- Note that \begin{align} \E\left(\left[Y - u(X)\right]^2\right) & = \E\left(\left[Y - \E(Y \mid X) + \E(Y \mid X) - u(X)\right]^2\right) \\ & = \E\left(\left[Y - \E(Y \mid X)\right]^2 \right) + 2 \E\left(\left[Y - \E(Y \mid X)\right] \left[\E(Y \mid X) - u(X)\right]\right) + \E\left(\left[\E(Y \mid X) - u(X)\right]^2\right) \end{align} But \( Y - \E(Y \mid X) \) has mean 0, so the middle term on the right is \( 2 \cov\left[Y - \E(Y \mid X), \E(Y \mid X) - u(X)\right] \). Moreover, \( \E(Y \mid X) - u(X) \) is a function of \( X \) and hence is uncorrelated with \( Y - \E(Y \mid X) \) by the general uncorrelated property . Hence the middle term is 0, so \[ \E\left(\left[Y - u(X)\right]^2\right) = \E\left(\left[Y - \E(Y \mid X)\right]^2 \right) + \E\left(\left[\E(Y \mid X) - u(X)\right]^2\right) \] and therefore \( \E\left(\left[Y - \E(Y \mid X)\right]^2 \right) \le \E\left(\left[Y - u(X)\right]^2\right) \).
- Equality holds if and only if \( \E\left(\left[\E(Y \mid X) - u(X)\right]^2\right) = 0 \), if and only if \( \P\left[u(X) = \E(Y \mid X)\right] = 1 \).
Suppose now that \(X\) is real-valued. In the section on covariance and correlation, we found that the best linear predictor of \(Y\) given \(X\) is
\[ L(Y \mid X) = \E(Y) + \frac{\cov(X,Y)}{\var(X)} \left[X - \E(X)\right] \]
On the other hand, \(\E(Y \mid X)\) is the best predictor of \(Y\) among all functions of \(X\). It follows that if \(\E(Y \mid X)\) happens to be a linear function of \(X\) then it must be the case that \(\E(Y \mid X) = L(Y \mid X)\). However, we will give a direct proof also:
If \(\E(Y \mid X) = a + b X\) for constants \(a\) and \(b\) then \( \E(Y \mid X) = L(Y \mid X) \); that is,
- \(b = \cov(X,Y) \big/ \var(X) \)
- \(a = \E(Y) - \E(X) \cov(X,Y) \big/ \var(X) \)
Proof
First, \( \E(Y) = \E\left[\E(Y \mid X)\right] = a + b \E(X) \), so \( a = \E(Y) - b \E(X) \). Next, \( \cov(X, Y) = \cov[X \E(Y \mid X)] = \cov(X, a + b X) = b \var(X) \) and therefore \( b = \cov(X, Y) \big/ \var(X) \).
Conditional Variance
The conditional variance of \( Y \) given \( X \) is defined like the ordinary variance, but with all expected values conditioned on \( X \).
The conditional variance of \(Y\) given \(X\) is defined as \[ \var(Y \mid X) = \E\left(\left[Y - \E(Y \mid X)\right]^2 \biggm| X \right) \]
Thus, \( \var(Y \mid X) \) is a function of \( X \), and in particular, is a random variable. Our first result is a computational formula that is analogous to the one for standard variance—the variance is the mean of the square minus the square of the mean, but now with all expected values conditioned on \( X \):
\(\var(Y \mid X) = \E\left(Y^2 \mid X\right) - \left[\E(Y \mid X)\right]^2\).
Proof
Expanding the square in the definition and using basic properties of conditional expectation, we have
\begin{align} \var(Y \mid X) & = \E\left(Y^2 - 2 Y \E(Y \mid X) + \left[\E(Y \mid X)\right]^2 \biggm| X \right) = \E(Y^2 \mid X) - 2 \E\left[Y \E(Y \mid X) \mid X\right] + \E\left(\left[\E(Y \mid X)\right]^2 \mid X\right) \\ & = \E\left(Y^2 \mid X\right) - 2 \E(Y \mid X) \E(Y \mid X) + \left[\E(Y \mid X)\right]^2 = \E\left(Y^2 \mid X\right) - \left[\E(Y \mid X)\right]^2 \end{align}Our next result shows how to compute the ordinary variance of \( Y \) by conditioning on \( X \) .
\(\var(Y) = \E\left[\var(Y \mid X)\right] + \var\left[\E(Y \mid X)\right]\).
Proof
From the previous theorem and properties of conditional expected value we have \( \E\left[\var(Y \mid X)\right] = \E\left(Y^2\right) - \E\left(\left[\E(Y \mid X)\right]^2\right) \). But \( \E\left(Y^2\right) = \var(Y) + \left[\E(Y)\right]^2 \) and similarly, \(\E\left(\left[\E(Y \mid X)\right]^2\right) = \var\left[\E(Y \mid X)\right] + \left(\E\left[\E(Y \mid X)\right]\right)^2 \). But also, \( \E\left[\E(Y \mid X)\right] = \E(Y) \) so subsituting we get \( \E\left[\var(Y \mid X)\right] = \var(Y) - \var\left[\E(Y \mid X)\right] \).
Thus, the variance of \( Y \) is the expected conditional variance plus the variance of the conditional expected value. This result is often a good way to compute \(\var(Y)\) when we know the conditional distribution of \(Y\) given \(X\). With the help of (21) we can give a formula for the mean square error when \(\E(Y \mid X)\) is used a predictor of \(Y\).
Mean square error \[ \E\left(\left[Y - \E(Y \mid X)\right]^2\right) = \var(Y) - \var\left[E(Y \mid X)\right] \]
Proof
From the definition of conditional variance , and using mean property and variance formula we have \[ \E\left(\left[Y - \E(Y \mid X)\right]^2\right) = \E\left[\var(Y \mid X)\right] = \var(Y) - \var\left[E(Y \mid X)\right] \]
Let us return to the study of predictors of the real-valued random variable \(Y\), and compare the three predictors we have studied in terms of mean square error.
Suppose that \( Y \) is a real-valued random variable.
- The best constant predictor of \(Y\) is \(\E(Y)\) with mean square error \(\var(Y)\).
- If \(X\) is another real-valued random variable, then the best linear predictor of \(Y\) given \(X\) is \[ L(Y \mid X) = \E(Y) + \frac{\cov(X,Y)}{\var(X)}\left[X - \E(X)\right] \] with mean square error \(\var(Y)\left[1 - \cor^2(X,Y)\right]\).
- If \(X\) is a general random variable, then the best overall predictor of \(Y\) given \(X\) is \(\E(Y \mid X)\) with mean square error \(\var(Y) - \var\left[\E(Y \mid X)\right]\).
Conditional Covariance
Suppose that \( Y \) and \( Z \) are real-valued random variables, and that \( X \) is a general random variable, all defined on our underlying probability space. Analogous to variance, the conditional covariance of \( Y \) and \( Z \) given \( X \) is defined like the ordinary covariance, but with all expected values conditioned on \( X \).
The conditional covariance of \(Y\) and \( Z \) given \(X\) is defined as \[ \cov(Y, Z \mid X) = \E\left([Y - \E(Y \mid X)] [Z - \E(Z \mid X) \biggm| X \right) \]
Thus, \( \cov(Y, Z \mid X) \) is a function of \( X \), and in particular, is a random variable. Our first result is a computational formula that is analogous to the one for standard covariance—the covariance is the mean of the product minus the product of the means, but now with all expected values conditioned on \( X \):
\(\cov(Y, Z \mid X) = \E\left(Y Z \mid X\right) - \E(Y \mid X) E(Z \mid X)\).
Proof
Expanding the product in the definition and using basic properties of conditional expectation, we have
\begin{align} \cov(Y, Z \mid X) & = \E\left(Y Z - Y \E(Z \mid X) - Z E(Y \mid X) + \E(Y \mid X) E(Z \mid X) \biggm| X \right) = \E(Y Z \mid X) - \E\left[Y \E(Z \mid X) \mid X\right] - \E\left[Z \E(Y \mid X) \mid X\right] + \E\left[\E(Y \mid X) \E(Z \mid X) \mid X\right] \\ & = \E\left(Y Z \mid X\right) - \E(Y \mid X) \E(Z \mid X) - \E(Y \mid X) \E(Z \mid X) + \E(Y \mid X) \E(Z \mid X) = \E\left(Y Z \mid X\right) - \E(Y \mid X) E(Z \mid X) \end{align}Our next result shows how to compute the ordinary covariance of \( Y \) and \( Z \) by conditioning on \( X \) .
\(\cov(Y, Z) = \E\left[\cov(Y, Z \mid X)\right] + \cov\left[\E(Y \mid X), \E(Z \mid X) \right]\).
Proof
From (25) and properties of conditional expected value we have \[ \E\left[\cov(Y, Z \mid X)\right] = \E(Y Z) - \E\left[\E(Y\mid X) \E(Z \mid X) \right] \] But \( \E(Y Z) = \cov(Y, Z) + \E(Y) \E(Z)\) and similarly, \[\E\left[\E(Y \mid X) \E(Z \mid X)\right] = \cov[\E(Y \mid X), \E(Z \mid X) + \E[\E(Y\mid X)] \E[\E(Z \mid X)]\] But also, \( \E[\E(Y \mid X)] = \E(Y) \) and \( \E[\E(Z \mid X)] = \E(Z) \) so subsituting we get \[ \E\left[\cov(Y, Z \mid X)\right] = \cov(Y, Z) - \cov\left[E(Y \mid X), E(Z \mid X)\right] \]
Thus, the covariance of \( Y \) and \( Z \) is the expected conditional covariance plus the covariance of the conditional expected values. This result is often a good way to compute \(\cov(Y, Z)\) when we know the conditional distribution of \((Y, Z)\) given \(X\).
Examples and Applications
As always, be sure to try the proofs and computations yourself before reading the ones in the text.
Simple Continuous Distributions
Suppose that \((X,Y)\) has probability density function \(f\) defined by \(f(x,y) = x + y\) for \(0 \le x \le 1\), \(0 \le y \le 1\).
- Find \(L(Y \mid X)\).
- Find \(\E(Y \mid X)\).
- Graph \(L(Y \mid X = x)\) and \(\E(Y \mid X = x)\) as functions of \(x\), on the same axes.
- Find \(\var(Y)\).
- Find \(\var(Y)\left[1 - \cor^2(X, Y)\right]\).
- Find \(\var(Y) - \var\left[\E(Y \mid X)\right]\).
Answer
- \(\frac{7}{11} - \frac{1}{11} X\)
- \(\frac{3 X + 2}{6 X + 3}\)
- \(\frac{11}{144} = 0.0764\)
- \(\frac{5}{66} = 0.0758\)
- \(\frac{1}{12} - \frac{1}{144} \ln 3 = 0.0757\)
Suppose that \((X,Y)\) has probability density function \(f\) defined by \(f(x,y) = 2 (x + y)\) for \(0 \le x \le y \le 1\).
- Find \(L(Y \mid X)\).
- Find \(\E(Y \mid X)\).
- Graph \(L(Y \mid X = x)\) and \(\E(Y \mid X = x)\) as functions of \(x\), on the same axes.
- Find \(\var(Y)\).
- Find \(\var(Y)\left[1 - \cor^2(X, Y)\right]\).
- Find \(\var(Y) - \var\left[\E(Y \mid X)\right]\).
Answer
- \(\frac{26}{43} + \frac{15}{43} X\)
- \(\frac{5 X^2 + 5 X + 2}{9 X + 3}\)
- \(\frac{3}{80} = 0.0375\)
- \(\frac{13}{430} = 0.0302\)
- \(\frac{1837}{21\;870} - \frac{512}{6561} \ln(2) = 0.0299\)
Suppose that \((X,Y)\) has probability density function \(f\) defined by \(f(x,y) = 6 x^2 y\) for \(0 \le x \le 1\), \(0 \le y \le 1\).
- Find \(L(Y \mid X)\).
- Find \(\E(Y \mid X)\).
- Graph \(L(Y \mid X = x)\) and \(\E(Y \mid X = x)\) as functions of \(x\), on the same axes.
- Find \(\var(Y)\).
- Find \(\var(Y)\left[1 - \cor^2(X, Y)\right]\).
- Find \(\var(Y) - \var\left[\E(Y \mid X)\right]\).
Answer
Note that \(X\) and \(Y\) are independent.
- \(\frac{2}{3}\)
- \(\frac{2}{3}\)
- \(\frac{1}{18}\)
- \(\frac{1}{18}\)
- \(\frac{1}{18}\)
Suppose that \((X,Y)\) has probability density function \(f\) defined by \(f(x,y) = 15 x^2 y\) for \(0 \le x \le y \le 1\).
- Find \(L(Y \mid X)\).
- Find \(\E(Y \mid X)\).
- Graph \(L(Y \mid X = x)\) and \(\E(Y \mid X = x)\) as functions of \(x\), on the same axes.
- Find \(\var(Y)\).
- Find \(\var(Y)\left[1 - \cor^2(X, Y)\right]\).
- Find \(\var(Y) - \var\left[\E(Y \mid X)\right]\).
Answer
- \(\frac{30}{51} + \frac{20}{51}X\)
- \(\frac{2(X^2 + X + 1)}{3(X + 1)}\)
- \(\frac{5}{252} = 0.0198\)
- \(\frac{5}{357} = 0.0140\)
- \(\frac{292}{63} - \frac{20}{3} \ln(2) = 0.0139\)
Exercises on Basic Properties
Suppose that \(X\), \(Y\), and \(Z\) are real-valued random variables with \(\E(Y \mid X) = X^3\) and \(\E(Z \mid X) = \frac{1}{1 + X^2}\). Find \(\E\left(Y\,e^X - Z\,\sin X \mid X\right)\).
Answer
\(X^3 e^X - \frac{\sin X}{1 + X^2}\)
Uniform Distributions
As usual, continuous uniform distributions can give us some geometric insight.
Recall first that for \( n \in \N_+ \), the standard measure on \(\R^n\) is \[\lambda_n(A) = \int_A 1 dx, \quad A \subseteq \R^n\] In particular, \(\lambda_1(A)\) is the length of \(A \subseteq \R\), \(\lambda_2(A)\) is the area of \(A \subseteq \R^2\), and \(\lambda_3(A)\) is the volume of \(A \subseteq \R^3\).
Details
Technically \(\lambda_n\) is Lebesgue measure on the measurable subsets of \(\R^n\). The integral representation is valid for the types of sets that occur in applications. In the discussion below, all subsets are assumed to be measurable.
With our usual setup, suppose that \(X\) takes values in \(S \subseteq \R^n\), \(Y\) takes values in \(T \subseteq \R\), and that \((X, Y)\) is uniformly distributed on \(R \subseteq S \times T \subseteq \R^{n+1}\). So \(0 \lt \lambda_{n+1}(R) \lt \infty\), and the joint probability density function \(f\) of \((X, Y)\) is given by \(f(x, y) = 1 / \lambda_{n+1}(R)\) for \((x, y) \in R\). Recall that uniform distributions, whether discrete or continuous, always have constant densities. Finally, recall that the cross section of \(R\) at \(x \in S\) is \(T_x = \{y \in T: (x, y) \in R\}\).
In the setting above, suppose that \( T_x \) is a bounded interval with midpoint \( m(x) \) and length \( l(x) \) for each \( x \in S \). Then
- \( \E(Y \mid X) = m(X) \)
- \( \var(Y \mid X) = \frac{1}{12}l^2(X) \)
Proof
This follows immediately from the fact that the conditional distribution of \( Y \) given \( X = x \) is uniformly distributed on \( T_x \) for each \( x \in S \).
So in particular, the regression curve \(x \mapsto \E(Y \mid X = x)\) follows the midpoints of the cross-sectional intervals.
In each case below, suppose that \( (X,Y) \) is uniformly distributed on the give region. Find \(\E(Y \mid X)\) and \( \var(Y \mid X) \)
- The rectangular region \(R = [a, b] \times [c, d]\) where \(a \lt b\) and \(c \lt d\).
- The triangular region \(T = \left\{(x,y) \in \R^2: -a \le x \le y \le a\right\}\) where \(a \gt 0\).
- The circular region \( C = \left\{(x, y) \in \R^2: x^2 + y^2 \le r\right\} \) where \( r \gt 0 \).
Answer
- \(\E(Y \mid X) = \frac{1}{2}(c + d)\), \( \var(Y \mid X) = \frac{1}{12}(d - c)^2 \). Note that \( X \) and \( Y \) are independent.
- \(\E(Y \mid X) = \frac{1}{2}(a + X)\), \( \var(Y \mid X) = \frac{1}{12}(a - X)^2 \)
- \( \E(Y \mid X) = 0 \), \( \var(Y \mid X) = 4 (r^2 - X^2) \)
In the bivariate uniform experiment, select each of the following regions. In each case, run the simulation 2000 times and note the relationship between the cloud of points and the graph of the regression function.
- square
- triangle
- circle
Suppose that \(X\) is uniformly distributed on the interval \((0, 1)\), and that given \(X\), random variable \(Y\) is uniformly distributed on \((0, X)\). Find each of the following:
- \(\E(Y \mid X)\)
- \(\E(Y)\)
- \(\var(Y \mid X)\)
- \(\var(Y)\)
Answer
- \(\frac{1}{2} X\)
- \(\frac{1}{4}\)
- \(\frac{1}{12} X^2\)
- \(\frac{7}{144}\)
The Hypergeometric Distribution
Suppose that a population consists of \(m\) objects, and that each object is one of three types. There are \(a\) objects of type 1, \(b\) objects of type 2, and \(m - a - b\) objects of type 0. The parameters \(a\) and \(b\) are positive integers with \(a + b \lt m\). We sample \(n\) objects from the population at random, and without replacement, where \( n \in \{0, 1, \ldots, m\} \). Denote the number of type 1 and 2 objects in the sample by \(X\) and \(Y\), so that the number of type 0 objects in the sample is \(n - X - Y\). In the in the chapter on Distributions, we showed that the joint, marginal, and conditional distributions of \( X \) and \( Y \) are all hypergeometric—only the parameters change. Here is the relevant result for this section:
In the setting above,
- \( \E(Y \mid X) = \frac{b}{m - a}(n - X) \)
- \( \var(Y \mid X) = \frac{b (m - a - b)}{(m - a)^2 (m - a - 1)} (n - X) (m - a - n + X)\)
- \( \E\left([Y - \E(Y \mid X)]^2\right) = \frac{n(m - n)b(m - a - b)}{m (m - 1)(m - a)} \)
Proof
Recall that \( (X, Y) \) has the (multivariate) hypergeometric distribution with parameters \( m \), \( a \), \( b \), and \( n \). Marginally, \( X \) has the hypergeometric distribution with parameters \( m \), \( a \), and \( n \), and \( Y \) has the hypergeometric distribution with parameters \( m \), \( b \), and \( n \). Given \( X = x \in \{0, 1, \ldots, n\} \), the remaining \( n - x \) objects are chosen at random from a population of \( m - a \) objects, of which \( b \) are type 2 and \( m - a - b \) are type 0. Hence, the conditional distribution of \( Y \) given \( X = x \) is hypergeometric with parameters \( m - a \), \( b \), and \( n - x \). Parts (a) and (b) then follow from the standard formulas for the mean and variance of the hypergeometric distribution, as functions of the parameters. Part (c) is the mean square error, and in this case can be computed most easily as \[ \var(Y) - \var[\E(Y \mid X)] = \var(Y) - \left(\frac{b}{m - a}\right)^2 \var(X) = n \frac{b}{m} \frac{m - b}{m} \frac{m - n}{m - 1} - \left(\frac{b}{m - a}\right)^2 n \frac{a}{m} \frac{m - a}{m} \frac{m - n}{m - 1} \] Simplifying gives the result.
Note that \( \E(Y \mid X) \) is a linear function of \( X \) and hence \( \E(Y \mid X) = L(Y \mid X) \).
In a collection of 120 objects, 50 are classified as good , 40 as fair and 30 as poor . A sample of 20 objects is selected at random and without replacement. Let \( X \) denote the number of good objects in the sample and \( Y \) the number of poor objects in the sample. Find each of the following:
- \( \E(Y \mid X) \)
- \( \var(Y \mid X) \)
- The predicted value of \( Y \) given \( X = 8 \)
Answer
- \( \E(Y \mid X) = \frac{80}{7} - \frac{4}{7} X \)
- \( \var(Y \mid X) = \frac{4}{1127}(20 - X)(50 + X) \)
- \( \frac{48}{7} \)
The Multinomial Trials Model
Suppose that we have a sequence of \( n \) independent trials, and that each trial results in one of three outcomes, denoted 0, 1, and 2. On each trial, the probability of outcome 1 is \( p \), the probability of outcome 2 is \( q \), so that the probability of outcome 0 is \( 1 - p - q \). The parameters \( p, \, q \in (0, 1) \) with \( p + q \lt 1 \), and of course \( n \in \N_+ \). Let \( X \) denote the number of trials that resulted in outcome 1, \( Y \) the number of trials that resulted in outcome 2, so that \( n - X - Y \) is the number of trials that resulted in outcome 0. In the in the chapter on Distributions, we showed that the joint, marginal, and conditional distributions of \( X \) and \( Y \) are all multinomial—only the parameters change. Here is the relevant result for this section:
In the setting above,
- \( \E(Y \mid X) = \frac{q}{1 - p}(n - X) \)
- \( \var(Y \mid X) = \frac{q (1 - p - q)}{(1 - p)^2}(n - X)\)
- \( \E\left([Y - \E(Y \mid X)]^2\right) = \frac{q (1 - p - q)}{1 - p} n \)
Proof
Recall that \( (X, Y) \) has the multinomial distribution with parameters \( n \), \( p \), and \( q \). Marginally, \( X \) has the binomial distribution with parameters \( n \) and \( p \), and \( Y \) has the binomial distribution with parameters \( n \) and \( q \). Given \( X = x \in \{0, 1, \ldots, n\} \), the remaining \( n - x \) trials are independent, but with just two outcomes: outcome 2 occurs with probability \( q / (1 - p) \) and outcome 0 occurs with probability \( 1 - q / (1 - p) \). (These are the conditional probabilities of outcomes 2 and 0, respectively, given that outcome 1 did not occur.) Hence the conditional distribution of \( Y \) given \( X = x \) is binomial with parameters \( n - x \) and \( q / (1 - p) \). Parts (a) and (b) then follow from the standard formulas for the mean and variance of the binomial distribution, as functions of the parameters. Part (c) is the mean square error and in this case can be computed most easily from \[ \E[\var(Y \mid X)] = \frac{q (1 - p - q)}{(1 - p)^2} [n - \E(X)] = \frac{q (1 - p - q)}{(1 - p)^2} (n - n p) = \frac{q (1 - p - q)}{1 - p} n\]
Note again that \( \E(Y \mid X) \) is a linear function of \( X \) and hence \( \E(Y \mid X) = L(Y \mid X) \).
Suppose that a fair, 12-sided die is thrown 50 times. Let \( X \) denote the number of throws that resulted in a number from 1 to 5, and \( Y \) the number of throws that resulted in a number from 6 to 9. Find each of the following:
- \( \E(Y \mid X) \)
- \( \var(Y \mid X) \)
- The predicted value of \( Y \) given \( X = 20 \)
Answer
- \( \E(Y \mid X) = \frac{4}{7}(50 - X) \)
- \( \var(Y \mid X) = \frac{12}{49}(50 - X) \)
- \( \frac{120}{7} \)
The Poisson Distribution
Recall that the
Poisson distribution
, named for Simeon Poisson, is widely used to model the number of
random points
in a region of time or space, under certain ideal conditions. The Poisson distribution is studied in more detail in the chapter on the Poisson Process. The Poisson distribution with parameter \( r \in (0, \infty) \) has probability density function \(f\) defined by \[ f(x) = e^{-r} \frac{r^x}{x!}, \quad x \in \N \] The parameter \( r \) is the mean and variance of the distribution.
Suppose that \( X \) and \( Y \) are independent random variables, and that \( X \) has the Poisson distribution with parameter \( a \in (0, \infty) \) and \( Y \) has the Poisson distribution with parameter \( b \in (0, \infty) \). Let \( N = X + Y \). Then
- \( \E(X \mid N) = \frac{a}{a + b}N\)
- \( \var(X \mid N) = \frac{a b}{(a + b)^2} N \)
- \( \E\left([X - \E(X \mid N)]^2\right) = \frac{a b}{a + b} \)
Proof
We have shown before that the distribution of \( N \) is also Poisson, with parameter \( a + b \), and that the conditional distribution of \( X \) given \( N = n \in \N \) is binomial with parameters \( n \) and \( a / (a + b) \). Hence parts (a) and (b) follow from the standard formulas for the mean and variance of the binomial distribution, as functions of the parameters. Part (c) is the mean square error, and in this case can be computed most easily as \[ \E[\var(X \mid N)] = \frac{a b}{(a + b)^2} \E(N) = \frac{ab}{(a + b)^2} (a + b) = \frac{a b}{a + b} \]
Once again, \( \E(X \mid N) \) is a linear function of \( N \) and so \( \E(X \mid N) = L(X \mid N) \). If we reverse the roles of the variables, the conditional expected value is trivial from our basic properties: \[ \E(N \mid X) = \E(X + Y \mid X) = X + b \]
Coins and Dice
A pair of fair dice are thrown, and the scores \((X_1, X_2)\) recorded. Let \(Y = X_1 + X_2\) denote the sum of the scores and \(U = \min\left\{X_1, X_2\right\}\) the minimum score. Find each of the following:
- \(\E\left(Y \mid X_1\right)\)
- \(\E\left(U \mid X_1\right)\)
- \(\E\left(Y \mid U\right)\)
- \(\E\left(X_2 \mid X_1\right)\)
Answer
- \(\frac{7}{2} + X_1\)
-
\(x\) 1 2 3 4 5 6 \(\E(U \mid X_1 = x)\) 1 \(\frac{11}{6}\) \(\frac{5}{2}\) 3 \(\frac{10}{3}\) \(\frac{7}{2}\) -
\(u\) 1 2 3 4 5 6 \(\E(Y \mid U = u)\) \(\frac{52}{11}\) \(\frac{56}{9}\) \(\frac{54}{7}\) \(\frac{46}{5}\) \(\frac{32}{3}\) 12 - \(\frac{7}{2}\)
A box contains 10 coins, labeled 0 to 9. The probability of heads for coin \(i\) is \(\frac{i}{9}\). A coin is chosen at random from the box and tossed. Find the probability of heads.
Answer
\(\frac{1}{2}\)
This problem is an example of Laplace's rule of succession, named for Pierre Simon Laplace.
Random Sums of Random Variables
Suppose that \(\bs{X} = (X_1, X_2, \ldots)\) is a sequence of independent and identically distributed real-valued random variables. We will denote the common mean, variance, and moment generating function, respectively, by \(\mu = \E(X_i)\), \(\sigma^2 = \var(X_i)\), and \(G(t) = \E\left(e^{t\,X_i}\right)\). Let \[ Y_n = \sum_{i=1}^n X_i, \quad n \in \N \] so that \((Y_0, Y_1, \ldots)\) is the partial sum process associated with \(\bs{X}\). Suppose now that \(N\) is a random variable taking values in \(\N\), independent of \(\bs{X}\). Then \[ Y_N = \sum_{i=1}^N X_i \] is a random sum of random variables; the terms in the sum are random, and the number of terms is random. This type of variable occurs in many different contexts. For example, \(N\) might represent the number of customers who enter a store in a given period of time, and \(X_i\) the amount spent by the customer \(i\), so that \( Y_N \) is the total revenue of the store during the period.
The conditional and ordinary expected value of \(Y_N\) are
- \(\E\left(Y_N \mid N\right) = N \mu\)
- \(\E\left(Y_N\right) = \E(N) \mu\)
Proof
- Using the substitution rule and the independence of \( N \) and \( \bs{X} \) we have \[ \E\left(Y_N \mid N = n\right) = \E\left(Y_n \mid N = n\right) = \E(Y_n) = \sum_{i=1}^n \E(X_i) = n \mu \] so \(\E\left(Y_N \mid N\right) = N \mu\).
- From (a) and conditioning, \( E\left(Y_N\right) = \E\left[\E\left(Y_N \mid N\right)\right] = \E(N \mu) = \E(N) \mu \).
Wald's equation , named for Abraham Wald, is a generalization of the previous result to the case where \(N\) is not necessarily independent of \(\bs{X}\), but rather is a stopping time for \(\bs{X}\). Roughly, this means that the event \( N = n \) depends only \( (X_1, X_2, \ldots, X_n) \). Wald's equation is discussed in the chapter on Random Samples. An elegant proof of and Wald's equation is given in the chapter on Martingales. The advanced section on stopping times is in the chapter on Probability Measures.
The conditional and ordinary variance of \(Y_N\) are
- \(\var\left(Y_N \mid N\right) = N \sigma^2\)
- \(\var\left(Y_N\right) = \E(N) \sigma^2 + \var(N) \mu^2\)
Proof
- Using the substitution rule, the independence of \( N \) and \( \bs{X} \), and the fact that \( \bs{X} \) is an IID sequence, we have \[ \var\left(Y_N \mid N = n\right) = \var\left(Y_n \mid N = n\right) = \var\left(Y_n\right) = \sum_{i=1}^n \var(X_i) = n \sigma^2 \] so \( \var\left(Y_N \mid N\right) = N \sigma^2 \).
- From (a) and the previous result, \[ \var\left(Y_N\right) = \E\left[\var\left(Y_N \mid N\right)\right] + \var\left[\E(Y_N \mid \N)\right] = \E(\sigma^2 N) + \var(\mu N) = \E(N) \sigma^2 + \mu^2 \var(N)\]
Let \(H\) denote the probability generating function of \(N\). The conditional and ordinary moment generating function of \(Y_N\) are
- \(\E\left(e^{t Y_N} \mid N\right) = \left[G(t)\right]^N\)
- \(\E\left(e^{t N}\right) = H\left(G(t)\right)\)
Proof
- Using the substitution rule, the independence of \( N \) and \( \bs{X} \), and the fact that \( \bs{X} \) is an IID sequence, we have \[ \E\left(e^{t Y_N} \mid N = n\right) = \E\left(e^{t Y_n} \mid N = n\right) = \E\left(e^{t Y_n}\right) = \left[G(t)\right]^n \] (Recall that the MGF of the sum of independent variables is the product of the individual MGFs.)
- From (a) and conditioning, \( \E\left(e^{t N}\right) = \E\left[\E\left(e^{t N} \mid N\right)\right] = \E\left(G(t)^N\right) = H(G(t)) \).
Thus the moment generating function of \( Y_N \) is \( H \circ G \), the composition of the probability generating function of \( N \) with the common moment generating function of \( \bs{X} \), a simple and elegant result.
 Let \(N\) denote the die score and \(Y\) the number of heads. Find each of the following:
- The conditional distribution of \(Y\) given \(N\).
- \(\E\left(Y \mid N\right)\)
- \(\var\left(Y \mid N\right)\)
- \(\E\left(Y_i\right)\)
- \(\var(Y)\)
Answer
- Binomial with parameters \(N\) and \(p = \frac{1}{2}\)
- \(\frac{1}{2} N\)
- \(\frac{1}{4} N\)
- \(\frac{7}{4}\)
- \(\frac{7}{3}\)
Run the die-coin experiment 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation.
The number of customers entering a store in a given hour is a random variable with mean 20 and standard deviation 3. Each customer, independently of the others, spends a random amount of money with mean $50 and standard deviation $5. Find the mean and standard deviation of the amount of money spent during the hour.
Answer
- \($1000\)
- \($30.82\)
A coin has a random probability of heads \(V\) and is tossed a random number of times \(N\). Suppose that \(V\) is uniformly distributed on \([0, 1]\); \(N\) has the Poisson distribution with parameter \(a \gt 0\); and \(V\) and \(N\) are independent. Let \(Y\) denote the number of heads. Compute the following:
- \(\E(Y \mid N, V)\)
- \(\E(Y \mid N)\)
- \(\E(Y \mid V)\)
- \(\E(Y)\)
- \(\var(Y \mid N, V)\)
- \(\var(Y)\)
Answer
- \(N V\)
- \(\frac{1}{2} N\)
- \(a V\)
- \(\frac{1}{2} a\)
- \(N V (1 - V)\)
- \(\frac{1}{12} a^2 + \frac{1}{2} a\)
Mixtures of Distributions
Suppose that \(\bs{X} = (X_1, X_2, \ldots)\) is a sequence of real-valued random variables. Denote the mean, variance, and moment generating function of \( X_i \) by \(\mu_i = \E(X_i)\), \(\sigma_i^2 = \var(X_i)\), and \(M_i(t) = \E\left(e^{t\,X_i}\right)\), for \(i \in \N_+\). Suppose also that \(N\) is a random variable taking values in \(\N_+\), independent of \(\bs{X}\). Denote the probability density function of \(N\) by \(p_n = \P(N = n)\) for \(n \in \N_+\). The distribution of the random variable \(X_N\) is a mixture of the distributions of \(\bs{X} = (X_1, X_2, \ldots)\), with the distribution of \(N\) as the mixing distribution.
The conditional and ordinary expected value of \( X_N \) are
- \(\E(X_N \mid N) = \mu_N\)
- \(\E(X_N) = \sum_{n=1}^\infty p_n\,\mu_n\)
Proof
- Using the substitution rule and the independence of \( N \) and \( \bs{X} \), we have \( \E(X_N \mid N = n) = \E(X_n \mid N = n) = \E(X_n) = \mu_n \)
- From (a) and the conditioning rule, \[ \E\left(X_N\right) = \E\left[\E\left(X_N\right)\right] = \E\left(\mu_N\right) = \sum_{n=1}^\infty p_n \mu_n\]
The conditional and ordinary variance of \( X_N \) are
- \(\var\left(X_N \mid N\right) = \sigma_N^2\)
- \(\var(X_N) = \sum_{n=1}^\infty p_n (\sigma_n^2 + \mu_n^2) - \left(\sum_{n=1}^\infty p_n\,\mu_n\right)^2\).
Proof
- Using the substitution rule and the independence of \( N \) and \(\bs{X}\), we have \( \var\left(X_N \mid N = n\right) = \var\left(X_n \mid N = n\right) = \var\left(X_n\right) = \sigma_n^2 \)
- From (a) we have \begin{align} \var\left(X_N\right) & = \E\left[\var\left(X_N \mid N\right)\right] + \var\left[\E\left(X_N \mid N\right)\right] = \E\left(\sigma_N^2\right) + \var\left(\mu_N\right) = \E\left(\sigma_N^2\right) + \E\left(\mu_N^2\right) - \left[\E\left(\mu_N\right)\right]^2 \\ & = \sum_{n=1}^\infty p_n \sigma_n^2 + \sum_{n=1}^\infty p_n \mu_n^2 - \left(\sum_{n=1}^\infty p_n \mu_n\right)^2 \end{align}
The conditional and ordinary moment generating function of \( X_N \) are
- \( \E\left(e^{t X_N} \mid N\right) = M_N(t) \)
- \(\E\left(e^{tX_N}\right) = \sum_{i=1}^\infty p_i M_i(t)\).
Proof
- Using the substitution rule and the independence of \( N \) and \( \bs{X} \), we have \( \E\left(e^{t X_N} \mid N = n\right) = \E\left(e^{t X_n} \mid N = n\right) = \E\left(^{t X_n}\right) = M_n(t) \)
- From (a) and the conditioning rule, \( \E\left(e^{t X_N}\right) = \E\left[\E\left(e^{t X_N} \mid N\right)\right] = \E\left[M_N(t)\right] = \sum_{n=1}^\infty p_n M_n(t)\)
In the coin-die experiment, a biased coin is tossed with probability of heads \(\frac{1}{3}\). If the coin lands tails, a fair die is rolled; if the coin lands heads, an ace-six flat die is rolled (faces 1 and 6 have probability \(\frac{1}{4}\) each, and faces 2, 3, 4, 5 have probability \(\frac{1}{8}\) each). Find the mean and standard deviation of the die score.
Answer
- \(\frac{7}{2}\)
- \(1.8634\)
Run the coin-die experiment 1000 times and note the apparent convergence of the empirical mean and standard deviation to the distribution mean and standard deviation.


5.4F: Excretion and Absorption
The integumentary system functions in absorption (oxygen and some medications) and excretion (e.g., perspiration via the eccrine glands).
LEARNING OBJECTIVE
Describe the role of glands in excretion and absorption
Key Takeaways
Key Points
- Eccrine glands, the major sweat glands of the human body, produce a clear, odorless substance, consisting primarily of water and NaCl. NaCl is reabsorbed in the duct to reduce salt loss.
- Apocrine sweat glands are found only in certain locations of the body: the axillae (armpits), areola and nipples of the breast, ear canal, perianal region, and some parts of the external genitalia.
- The sebaceous glands secrete an oily/waxy matter called sebum to lubricate and waterproof the skin and hair of mammals. In humans, they are found in greatest abundance on the face and scalp, though they are distributed throughout all skin sites except the palms and soles.
A major function of the integumentary system is absorption and excretion.
Excretion
There are numerous secretory glands present in the skin which secrete a large range of distinct fluids.
Perspiration, or sweating, is the production of fluids secreted by the sweat glands in the skin of mammals. Two types of sweat glands can be found in humans: eccrine glands and apocrine glands.
Eccrine glands are the major sweat glands of the human body, found in virtually all skin. They produce a clear, odorless substance consisting primarily of water and NaCl (note that the odor from sweat is due to bacterial activity on the secretions of the apocrine glands).
NaCl is reabsorbed in the duct to reduce salt loss. Eccrine glands are active in thermoregulation and are stimulated by the sympathetic nervous system.
Sweat gland: A sectional view of the skin (magnified), with the eccrine glands highlighted.
Apocrine sweat glands are inactive until they are stimulated by hormonal changes in puberty. Apocrine sweat glands are mainly thought to function as olfactory pheromones, chemicals important in attracting a potential mate. The stimulus for the secretion of apocrine sweat glands is adrenaline, which is a hormone carried in the blood.
The sebaceous glands are microscopic glands in the skin that secrete an oily/waxy matter, called sebum, to lubricate and waterproof the skin and hair of mammals. In humans, they are found in greatest abundance on the face and scalp, though they are distributed throughout all skin sites except the palms and soles. In the eyelids, meibomian sebaceous glands secrete a special type of sebum into tears.
Absorption
Due to the absorptive capabilities of skin, the cells comprising the outermost 0.25–0.40 mm of the skin can be supplied by external oxygen rather than via the underlying capillary network. Additionally certain medications can be administered through the skin.
The most common mechanism of administration through the skin is the use of ointments or an adhesive patch, such as the nicotine patch or iontophoresis. Iontophoresis, also called electromotive drug administration, is a technique that uses a small electric charge to deliver a medicine or other chemical through the skin.
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2.2: Scales of Measurement
As the previous section indicates, the outcome of a psychological measurement is called a variable. But not all variables are of the same qualitative type, and it’s very useful to understand what types there are. A very useful concept for distinguishing between different types of variables is what’s known as scales of measurement .
Nominal scale
A nominal scale variable (also referred to as a categorical variable) is one in which there is no particular relationship between the different possibilities: for these kinds of variables it doesn’t make any sense to say that one of them is “bigger’ or”better" than any other one, and it absolutely doesn’t make any sense to average them. The classic example for this is “eye colour”. Eyes can be blue, green and brown, among other possibilities, but none of them is any “better” than any other one. As a result, it would feel really weird to talk about an “average eye colour”. Similarly, gender is nominal too: male isn’t better or worse than female, neither does it make sense to try to talk about an “average gender”. In short, nominal scale variables are those for which the only thing you can say about the different possibilities is that they are different. That’s it.
Let’s take a slightly closer look at this. Suppose I was doing research on how people commute to and from work. One variable I would have to measure would be what kind of transportation people use to get to work. This “transport type” variable could have quite a few possible values, including: “train”, “bus”, “car”, “bicycle”, etc. For now, let’s suppose that these four are the only possibilities, and suppose that when I ask 100 people how they got to work today, and I get this:
| Transportation | Number of people |
| (1) Train | 12 |
| (2) Bus | 30 |
| (3) Car | 48 |
| (4) Bicycle | 10 |
So, what’s the average transportation type? Obviously, the answer here is that there isn’t one. It’s a silly question to ask. You can say that travel by car is the most popular method, and travel by train is the least popular method, but that’s about all. Similarly, notice that the order in which I list the options isn’t very interesting. I could have chosen to display the data like this
| Transportation | Number of people |
| (3) Car | 48 |
| (1) Train | 12 |
| (4) Bicycle | 10 |
| (2) Bus | 30 |
and nothing really changes.
Ordinal scale
Ordinal scale variables have a bit more structure than nominal scale variables, but not by a lot. An ordinal scale variable is one in which there is a natural, meaningful way to order the different possibilities, but you can’t do anything else. The usual example given of an ordinal variable is “finishing position in a race”. You can say that the person who finished first was faster than the person who finished second, but you don’t know how much faster. As a consequence we know that 1st > 2nd, and we know that 2nd > 3rd, but the difference between 1st and 2nd might be much larger than the difference between 2nd and 3rd.
Here’s an more psychologically interesting example. Suppose I’m interested in people’s attitudes to climate change, and I ask them to pick one of these four statements that most closely matches their beliefs:
- Temperatures are rising, because of human activity
- Temperatures are rising, but we don’t know why
- Temperatures are rising, but not because of humans
- Temperatures are not rising
Notice that these four statements actually do have a natural ordering, in terms of “the extent to which they agree with the current science”. Statement 1 is a close match, statement 2 is a reasonable match, statement 3 isn’t a very good match, and statement 4 is in strong opposition to the science. So, in terms of the thing I’m interested in (the extent to which people endorse the science), I can order the items as 1 > 2 > 3 > 4. Since this ordering exists, it would be very weird to list the options like this…
- Temperatures are rising, but not because of humans
- Temperatures are rising, because of human activity
- Temperatures are not rising
- Temperatures are rising, but we don’t know why
… because it seems to violate the natural “structure” to the question.
So, let’s suppose I asked 100 people these questions, and got the following answers:
| Response | Number |
| (1) Temperatures are rising, because of human activity | 51 |
| (2) Temperatures are rising, but we don’t know why | 20 |
| (3) Temperatures are rising, but not because of humans | 10 |
| (4) Temperatures are not rising | 19 |
When analysing these data, it seems quite reasonable to try to group (1), (2) and (3) together, and say that 81 of 100 people were willing to at least partially endorse the science. And it’s also quite reasonable to group (2), (3) and (4) together and say that 49 of 100 people registered at least some disagreement with the dominant scientific view. However, it would be entirely bizarre to try to group (1), (2) and (4) together and say that 90 of 100 people said… what? There’s nothing sensible that allows you to group those responses together at all.
That said, notice that while we can use the natural ordering of these items to construct sensible groupings, what we can’t do is average them. For instance, in my simple example here, the “average” response to the question is 1.97. If you can tell me what that means, I’d love to know. Because that sounds like gibberish to me!
Interval scale
In contrast to nominal and ordinal scale variables, interval scale and ratio scale variables are variables for which the numerical value is genuinely meaningful. In the case of interval scale variables, the differences between the numbers are interpretable, but the variable doesn’t have a “natural” zero value. A good example of an interval scale variable is measuring temperature in degrees celsius. For instance, if it was 15 o yesterday and 18∘ today, then the 3 o difference between the two is genuinely meaningful. Moreover, that 3 o difference is exactly the same as the 3 o difference between 7 o and 10 o . In short, addition and subtraction are meaningful for interval scale variables. 8
However, notice that the 0 o does not mean “no temperature at all”: it actually means “the temperature at which water freezes”, which is pretty arbitrary. As a consequence, it becomes pointless to try to multiply and divide temperatures. It is wrong to say that 20 o is twice as hot as 10 o , just as it is weird and meaningless to try to claim that 20 o is negative two times as hot as -10 o .
Again, lets look at a more psychological example. Suppose I’m interested in looking at how the attitudes of first-year university students have changed over time. Obviously, I’m going to want to record the year in which each student started. This is an interval scale variable. A student who started in 2003 did arrive 5 years before a student who started in 2008. However, it would be completely insane for me to divide 2008 by 2003 and say that the second student started “1.0024 times later” than the first one. That doesn’t make any sense at all.
Ratio scale
The fourth and final type of variable to consider is a ratio scale variable, in which zero really means zero, and it’s okay to multiply and divide. A good psychological example of a ratio scale variable is response time (RT). In a lot of tasks it’s very common to record the amount of time somebody takes to solve a problem or answer a question, because it’s an indicator of how difficult the task is. Suppose that Alan takes 2.3 seconds to respond to a question, whereas Ben takes 3.1 seconds. As with an interval scale variable, addition and subtraction are both meaningful here. Ben really did take 3.1 - 2.3 = 0.8 seconds longer than Alan did. However, notice that multiplication and division also make sense here too: Ben took 3.1 / 2.3 = 1.35 times as long as Alan did to answer the question. And the reason why you can do this is that, for a ratio scale variable such as RT, “zero seconds” really does mean “no time at all”.
Continuous versus discrete variables
There’s a second kind of distinction that you need to be aware of, regarding what types of variables you can run into. This is the distinction between continuous variables and discrete variables. The difference between these is as follows:
- A continuous variable is one in which, for any two values that you can think of, it’s always logically possible to have another value in between.
- A discrete variable is, in effect, a variable that isn’t continuous. For a discrete variable, it’s sometimes the case that there’s nothing in the middle.
These definitions probably seem a bit abstract, but they’re pretty simple once you see some examples. For instance, response time is continuous. If Alan takes 3.1 seconds and Ben takes 2.3 seconds to respond to a question, then it’s possible for Cameron’s response time to lie in between, by taking 3.0 seconds. And of course it would also be possible for David to take 3.031 seconds to respond, meaning that his RT would lie in between Cameron’s and Alan’s. And while in practice it might be impossible to measure RT that precisely, it’s certainly possible in principle. Because we can always find a new value for RT in between any two other ones, we say that RT is continuous.
Discrete variables occur when this rule is violated. For example, nominal scale variables are always discrete: there isn’t a type of transportation that falls “in between” trains and bicycles, not in the strict mathematical way that 2.3 falls in between 2 and 3. So transportation type is discrete. Similarly, ordinal scale variables are always discrete: although “2nd place” does fall between “1st place” and “3rd place”, there’s nothing that can logically fall in between “1st place” and “2nd place”. Interval scale and ratio scale variables can go either way. As we saw above, response time (a ratio scale variable) is continuous. Temperature in degrees celsius (an interval scale variable) is also continuous. However, the year you went to school (an interval scale variable) is discrete. There’s no year in between 2002 and 2003. The number of questions you get right on a true-or-false test (a ratio scale variable) is also discrete: since a true-or-false question doesn’t allow you to be “partially correct”, there’s nothing in between 5/10 and 6/10. Table 2.1 summarises the relationship between the scales of measurement and the discrete/continuity distinction. Cells with a tick mark correspond to things that are possible. I’m trying to hammer this point home, because (a) some textbooks get this wrong, and (b) people very often say things like “discrete variable” when they mean “nominal scale variable”. It’s very unfortunate.
Table 2.1: The relationship between the scales of measurement and the discrete/continuity distinction. Cells with a tick mark correspond to things that are possible.
| continuous | discrete | |
| nominal | ✓ | |
| ordinal | ✓ | |
| interval | ✓ | ✓ |
| ratio | ✓ | ✓ |
Some complexities
Okay, I know you’re going to be shocked to hear this, but … the real world is much messier than this little classification scheme suggests. Very few variables in real life actually fall into these nice neat categories, so you need to be kind of careful not to treat the scales of measurement as if they were hard and fast rules. It doesn’t work like that: they’re guidelines, intended to help you think about the situations in which you should treat different variables differently. Nothing more.
So let’s take a classic example, maybe the classic example, of a psychological measurement tool: the Likert scale . The humble Likert scale is the bread and butter tool of all survey design. You yourself have filled out hundreds, maybe thousands of them, and odds are you’ve even used one yourself. Suppose we have a survey question that looks like this:
Which of the following best describes your opinion of the statement that “all pirates are freaking awesome” …
and then the options presented to the participant are these:
(1) Strongly disagree
(2) Disagree
(3) Neither agree nor disagree
(4) Agree
(5) Strongly agree
This set of items is an example of a 5-point Likert scale: people are asked to choose among one of several (in this case 5) clearly ordered possibilities, generally with a verbal descriptor given in each case. However, it’s not necessary that all items be explicitly described. This is a perfectly good example of a 5-point Likert scale too:
(1) Strongly disagree
(2)
(3)
(4)
(5) Strongly agree
Likert scales are very handy, if somewhat limited, tools. The question is, what kind of variable are they?  They’re obviously not nominal scale, since the items are ordered; and they’re not ratio scale either, since there’s no natural zero.
But are they ordinal scale or interval scale? One argument says that we can’t really prove that the difference between “strongly agree” and “agree” is of the same size as the difference between “agree” and “neither agree nor disagree”. In fact, in everyday life it’s pretty obvious that they’re not the same at all. So this suggests that we ought to treat Likert scales as ordinal variables. On the other hand, in practice most participants do seem to take the whole “on a scale from 1 to 5” part fairly seriously, and they tend to act as if the differences between the five response options were fairly similar to one another. As a consequence, a lot of researchers treat Likert scale data as if it were interval scale. It’s not interval scale, but in practice it’s close enough that we usually think of it as being quasi-interval scale.


7: Global Civil Society
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Patterns of globalisation have challenged the exclusivity of states as actors in international affairs. Globalisation links distant communities and opens up spaces for new social actors. Among the non-state actors benefiting from this change are public-interest-orientated non-governmental actors, often known as civil society groups. Alongside the state, profit-orientated corporate actors (which we will explore in the next chapter) and international governmental organisations (which we explored in the previous chapter), these civil society groups complete the mosaic of actors on the international stage.
The standard definition of civil society identifies it as the space outside of government, family and market.  Civil society organisations can include community groups, non-governmental organisations, social movements, labour unions, indigenous groups, charitable organisations, faithbased organisations, media operators, academia, diaspora groups, lobby and consultancy groups, think tanks and research centres, professional associations, and foundations. Political parties and private companies can also be counted as borderline cases. The presence of civil society organisations in international affairs has become increasingly relevant. They have played a role in agenda setting, international law-making and diplomacy. Further, they have been involved in the implementation and monitoring of a number of crucial global issues. These range from trade to development and poverty reduction, from democratic governance to human rights, from peace to the environment, and from security to the information society. Because of these reasons, international relations cannot be fully captured without taking into account the actions of civil society organisations.
Different theoretical perspectives can be used to interpret global civil society. Liberals may understand it as the actor that provides a bottom-up contribution to the effectiveness and legitimacy of the international system as a whole. In essence, it is democracy in action as power is being held to account by the populace. Realists, however, may interpret global civil society as a tool used by the most powerful states to advance their ultimate interests abroad, often promoting and popularising ideas that are key to the national interest. Marxists may see global civil society as political vanguards that can spread a different world view that challenges the dominant order. Finally, some even argue that the concept of civil society as a sphere distinct from the family, state and market remains a Western concept that does not apply easily to societies where the boundaries between these spheres are more blurred. It is useful to keep these various perspectives in mind as you read through the chapter.


44.2: Biogeography
Skills to Develop
- Define biogeography
- List and describe abiotic factors that affect the global distribution of plant and animal species
- Compare the impact of abiotic forces on aquatic and terrestrial environments
- Summarize the affect of abiotic factors on net primary productivity
Many forces influence the communities of living organisms present in different parts of the biosphere (all of the parts of Earth inhabited by life). The biosphere extends into the atmosphere (several kilometers above Earth) and into the depths of the oceans. Despite its apparent vastness to an individual human, the biosphere occupies only a minute space when compared to the known universe. Many abiotic forces influence where life can exist and the types of organisms found in different parts of the biosphere. The abiotic factors influence the distribution of biomes : large areas of land with similar climate, flora, and fauna.
Biogeography
Biogeography is the study of the geographic distribution of living things and the abiotic factors that affect their distribution. Abiotic factors such as temperature and rainfall vary based mainly on latitude and elevation. As these abiotic factors change, the composition of plant and animal communities also changes. For example, if you were to begin a journey at the equator and walk north, you would notice gradual changes in plant communities. At the beginning of your journey, you would see tropical wet forests with broad-leaved evergreen trees, which are characteristic of plant communities found near the equator. As you continued to travel north, you would see these broad-leaved evergreen plants eventually give rise to seasonally dry forests with scattered trees. You would also begin to notice changes in temperature and moisture. At about 30 degrees north, these forests would give way to deserts, which are characterized by low precipitation.
Moving farther north, you would see that deserts are replaced by grasslands or prairies. Eventually, grasslands are replaced by deciduous temperate forests. These deciduous forests give way to the boreal forests found in the subarctic, the area south of the Arctic Circle. Finally, you would reach the Arctic tundra, which is found at the most northern latitudes. This trek north reveals gradual changes in both climate and the types of organisms that have adapted to environmental factors associated with ecosystems found at different latitudes. However, different ecosystems exist at the same latitude due in part to abiotic factors such as jet streams, the Gulf Stream, and ocean currents. If you were to hike up a mountain, the changes you would see in the vegetation would parallel those as you move to higher latitudes.
Ecologists who study biogeography examine patterns of species distribution. No species exists everywhere; for example, the Venus flytrap is endemic to a small area in North and South Carolina. An endemic species is one which is naturally found only in a specific geographic area that is usually restricted in size. Other species are generalists: species which live in a wide variety of geographic areas; the raccoon, for example, is native to most of North and Central America.
Species distribution patterns are based on biotic and abiotic factors and their influences during the very long periods of time required for species evolution; therefore, early studies of biogeography were closely linked to the emergence of evolutionary thinking in the eighteenth century. Some of the most distinctive assemblages of plants and animals occur in regions that have been physically separated for millions of years by geographic barriers. Biologists estimate that Australia, for example, has between 600,000 and 700,000 species of plants and animals. Approximately 3/4 of living plant and mammal species are endemic species found solely in Australia (Figure \(\PageIndex{1}\)).
Sometimes ecologists discover unique patterns of species distribution by determining where species are not found. Hawaii, for example, has no native land species of reptiles or amphibians, and has only one native terrestrial mammal, the hoary bat. Most of New Guinea, as another example, lacks placental mammals.
Link to Learning
Check out this video to observe a platypus swimming in its natural habitat in New South Wales, Australia.
Plants can be endemic or generalists: endemic plants are found only on specific regions of the Earth, while generalists are found on many regions. Isolated land masses—such as Australia, Hawaii, and Madagascar—often have large numbers of endemic plant species. Some of these plants are endangered due to human activity. The forest gardenia ( Gardenia brighamii ), for instance, is endemic to Hawaii; only an estimated 15–20 trees are thought to exist (Figure \(\PageIndex{2}\)).
Energy Sources
Energy from the sun is captured by green plants, algae, cyanobacteria, and photosynthetic protists. These organisms convert solar energy into the chemical energy needed by all living things. Light availability can be an important force directly affecting the evolution of adaptations in photosynthesizers. For instance, plants in the understory of a temperate forest are shaded when the trees above them in the canopy completely leaf out in the late spring. Not surprisingly, understory plants have adaptations to successfully capture available light. One such adaptation is the rapid growth of spring ephemeral plants such as the spring beauty (Figure \(\PageIndex{3}\)). These spring flowers achieve much of their growth and finish their life cycle (reproduce) early in the season before the trees in the canopy develop leaves.
In aquatic ecosystems, the availability of light may be limited because sunlight is absorbed by water, plants, suspended particles, and resident microorganisms. Toward the bottom of a lake, pond, or ocean, there is a zone that light cannot reach. Photosynthesis cannot take place there and, as a result, a number of adaptations have evolved that enable living things to survive without light. For instance, aquatic plants have photosynthetic tissue near the surface of the water; for example, think of the broad, floating leaves of a water lily—water lilies cannot survive without light. In environments such as hydrothermal vents, some bacteria extract energy from inorganic chemicals because there is no light for photosynthesis.
The availability of nutrients in aquatic systems is also an important aspect of energy or photosynthesis. Many organisms sink to the bottom of the ocean when they die in the open water; when this occurs, the energy found in that living organism is sequestered for some time unless ocean upwelling occurs. Ocean upwelling is the rising of deep ocean waters that occurs when prevailing winds blow along surface waters near a coastline (Figure \(\PageIndex{4}\)). As the wind pushes ocean waters offshore, water from the bottom of the ocean moves up to replace this water. As a result, the nutrients once contained in dead organisms become available for reuse by other living organisms.
In freshwater systems, the recycling of nutrients occurs in response to air temperature changes. The nutrients at the bottom of lakes are recycled twice each year: in the spring and fall turnover. The spring and fall turnover is a seasonal process that recycles nutrients and oxygen from the bottom of a freshwater ecosystem to the top of a body of water (Figure \(\PageIndex{5}\)). These turnovers are caused by the formation of a thermocline : a layer of water with a temperature that is significantly different from that of the surrounding layers. In wintertime, the surface of lakes found in many northern regions is frozen. However, the water under the ice is slightly warmer, and the water at the bottom of the lake is warmer yet at 4 °C to 5 °C (39.2 °F to 41 °F). Water is densest at 4 °C; therefore, the deepest water is also the densest. The deepest water is oxygen poor because the decomposition of organic material at the bottom of the lake uses up available oxygen that cannot be replaced by means of oxygen diffusion into the water due to the surface ice layer.
Art Connection
How might turnover in tropical lakes differ from turnover in lakes that exist in temperate regions?
In springtime, air temperatures increase and surface ice melts. When the temperature of the surface water begins to reach 4 °C, the water becomes heavier and sinks to the bottom. The water at the bottom of the lake is then displaced by the heavier surface water and, thus, rises to the top. As that water rises to the top, the sediments and nutrients from the lake bottom are brought along with it. During the summer months, the lake water stratifies, or forms layers, with the warmest water at the lake surface.
As air temperatures drop in the fall, the temperature of the lake water cools to 4 °C; therefore, this causes fall turnover as the heavy cold water sinks and displaces the water at the bottom. The oxygen-rich water at the surface of the lake then moves to the bottom of the lake, while the nutrients at the bottom of the lake rise to the surface (Figure \(\PageIndex{5}\)). During the winter, the oxygen at the bottom of the lake is used by decomposers and other organisms requiring oxygen, such as fish.
Temperature
Temperature affects the physiology of living things as well as the density and state of water. Temperature exerts an important influence on living things because few living things can survive at temperatures below 0 °C (32 °F) due to metabolic constraints. It is also rare for living things to survive at temperatures exceeding 45 °C (113 °F); this is a reflection of evolutionary response to typical temperatures. Enzymes are most efficient within a narrow and specific range of temperatures; enzyme degradation can occur at higher temperatures. Therefore, organisms either must maintain an internal temperature or they must inhabit an environment that will keep the body within a temperature range that supports metabolism. Some animals have adapted to enable their bodies to survive significant temperature fluctuations, such as seen in hibernation or reptilian torpor. Similarly, some bacteria are adapted to surviving in extremely hot temperatures such as geysers. Such bacteria are examples of extremophiles: organisms that thrive in extreme environments.
Temperature can limit the distribution of living things. Animals faced with temperature fluctuations may respond with adaptations, such as migration, in order to survive. Migration, the movement from one place to another, is an adaptation found in many animals, including many that inhabit seasonally cold climates. Migration solves problems related to temperature, locating food, and finding a mate. In migration, for instance, the Arctic Tern ( Sterna paradisaea ) makes a 40,000 km (24,000 mi) round trip flight each year between its feeding grounds in the southern hemisphere and its breeding grounds in the Arctic Ocean. Monarch butterflies ( Danaus plexippus ) live in the eastern United States in the warmer months and migrate to Mexico and the southern United States in the wintertime. Some species of mammals also make migratory forays. Reindeer ( Rangifer tarandus ) travel about 5,000 km (3,100 mi) each year to find food. Amphibians and reptiles are more limited in their distribution because they lack migratory ability. Not all animals that can migrate do so: migration carries risk and comes at a high energy cost.
Some animals hibernate or estivate to survive hostile temperatures. Hibernation enables animals to survive cold conditions, and estivation allows animals to survive the hostile conditions of a hot, dry climate. Animals that hibernate or estivate enter a state known as torpor: a condition in which their metabolic rate is significantly lowered. This enables the animal to wait until its environment better supports its survival. Some amphibians, such as the wood frog ( Rana sylvatica ), have an antifreeze-like chemical in their cells, which retains the cells’ integrity and prevents them from bursting.
Water
Water is required by all living things because it is critical for cellular processes. Since terrestrial organisms lose water to the environment by simple diffusion, they have evolved many adaptations to retain water.
- Plants have a number of interesting features on their leaves, such as leaf hairs and a waxy cuticle, that serve to decrease the rate of water loss via transpiration.
- Freshwater organisms are surrounded by water and are constantly in danger of having water rush into their cells because of osmosis. Many adaptations of organisms living in freshwater environments have evolved to ensure that solute concentrations in their bodies remain within appropriate levels. One such adaptation is the excretion of dilute urine.
- Marine organisms are surrounded by water with a higher solute concentration than the organism and, thus, are in danger of losing water to the environment because of osmosis. These organisms have morphological and physiological adaptations to retain water and release solutes into the environment. For example, Marine iguanas ( Amblyrhynchus cristatus ), sneeze out water vapor that is high in salt in order to maintain solute concentrations within an acceptable range while swimming in the ocean and eating marine plants.
Inorganic Nutrients and Soil
Inorganic nutrients, such as nitrogen and phosphorus, are important in the distribution and the abundance of living things. Plants obtain these inorganic nutrients from the soil when water moves into the plant through the roots. Therefore, soil structure (particle size of soil components), soil pH, and soil nutrient content play an important role in the distribution of plants. Animals obtain inorganic nutrients from the food they consume. Therefore, animal distributions are related to the distribution of what they eat. In some cases, animals will follow their food resource as it moves through the environment.
Other Aquatic Factors
Some abiotic factors, such as oxygen, are important in aquatic ecosystems as well as terrestrial environments. Terrestrial animals obtain oxygen from the air they breathe. Oxygen availability can be an issue for organisms living at very high elevations, however, where there are fewer molecules of oxygen in the air. In aquatic systems, the concentration of dissolved oxygen is related to water temperature and the speed at which the water moves. Cold water has more dissolved oxygen than warmer water. In addition, salinity, current, and tide can be important abiotic factors in aquatic ecosystems.
Other Terrestrial Factors
Wind can be an important abiotic factor because it influences the rate of evaporation and transpiration. The physical force of wind is also important because it can move soil, water, or other abiotic factors, as well as an ecosystem’s organisms.
Fire is another terrestrial factor that can be an important agent of disturbance in terrestrial ecosystems. Some organisms are adapted to fire and, thus, require the high heat associated with fire to complete a part of their life cycle. For example, the jack pine—a coniferous tree—requires heat from fire for its seed cones to open (Figure \(\PageIndex{6}\)). Through the burning of pine needles, fire adds nitrogen to the soil and limits competition by destroying undergrowth.
Abiotic Factors Influencing Plant Growth
Temperature and moisture are important influences on plant production (primary productivity) and the amount of organic matter available as food (net primary productivity). Net primary productivity is an estimation of all of the organic matter available as food; it is calculated as the total amount of carbon fixed per year minus the amount that is oxidized during cellular respiration. In terrestrial environments, net primary productivity is estimated by measuring the aboveground biomass per unit area, which is the total mass of living plants, excluding roots. This means that a large percentage of plant biomass which exists underground is not included in this measurement. Net primary productivity is an important variable when considering differences in biomes. Very productive biomes have a high level of aboveground biomass.
Annual biomass production is directly related to the abiotic components of the environment. Environments with the greatest amount of biomass have conditions in which photosynthesis, plant growth, and the resulting net primary productivity are optimized. The climate of these areas is warm and wet. Photosynthesis can proceed at a high rate, enzymes can work most efficiently, and stomata can remain open without the risk of excessive transpiration; together, these factors lead to the maximal amount of carbon dioxide (CO 2 ) moving into the plant, resulting in high biomass production. The aboveground biomass produces several important resources for other living things, including habitat and food. Conversely, dry and cold environments have lower photosynthetic rates and therefore less biomass. The animal communities living there will also be affected by the decrease in available food.
Summary
Biogeography is the study of the geographic distribution of living things and the abiotic factors that affect their distribution. Endemic species are species that are naturally found only in a specific geographic area. The distribution of living things is influenced by several environmental factors that are, in part, controlled by the latitude or elevation at which an organism is found. Ocean upwelling and spring and fall turnovers are important processes regulating the distribution of nutrients and other abiotic factors important in aquatic ecosystems. Energy sources, temperature, water, inorganic nutrients, and soil are factors limiting the distribution of living things in terrestrial systems. Net primary productivity is a measure of the amount of biomass produced by a biome.
Art Connections
Figure \(\PageIndex{5}\): How might turnover in tropical lakes differ from turnover in lakes that exist in temperate regions?
- Answer
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Tropical lakes don’t freeze, so they don’t undergo spring turnover in the same way temperate lakes do. However, stratification does occur, as well as seasonal turnover.
Glossary
- aboveground biomass
- total mass of aboveground living plants per area
- biogeography
- study of the geographic distribution of living things and the abiotic factors that affect their distribution
- biome
- ecological community of plants, animals, and other organisms that is adapted to a characteristic set of environmental conditions
- endemic
- species found only in a specific geographic area that is usually restricted in size
- fall and spring turnover
- seasonal process that recycles nutrients and oxygen from the bottom of a freshwater ecosystem to the top
- net primary productivity
- measurement of the energy accumulation within an ecosystem, calculated as the total amount of carbon fixed per year minus the amount that is oxidized during cellular respiration
- ocean upwelling
- rising of deep ocean waters that occurs when prevailing winds blow along surface waters near a coastline
- thermocline
- layer of water with a temperature that is significantly different from that of the surrounding layers