6.4: The Central Limit Theorem
The central limit theorem and the law of large numbers are the two fundamental theorems of probability. Roughly, the central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution. The importance of the central limit theorem is hard to overstate; indeed it is the reason that many statistical procedures work.
Partial Sum Processes
Definitions
Suppose that \(\bs{X} = (X_1, X_2, \ldots)\) is a sequence of independent, identically distributed, real-valued random variables with common probability density function \(f\), mean \(\mu\), and variance \(\sigma^2\). We assume that \(0 \lt \sigma \lt \infty\), so that in particular, the random variables really are random and not constants. Let \[ Y_n = \sum_{i=1}^n X_i, \quad n \in \N \] Note that by convention, \(Y_0 = 0\), since the sum is over an empty index set. The random process \(\bs{Y} = (Y_0, Y_1, Y_2, \ldots)\) is called the partial sum process associated with \(\bs{X}\). Special types of partial sum processes have been studied in many places in this text; in particular see
- the binomial distribution in the setting of Bernoulli trials
- the negative binomial distribution in the setting of Bernoulli trials
- the gamma distribution in the Poisson process
- the the arrival times in a general renewal process
Recall that in statistical terms, the sequence \(\bs{X}\) corresponds to sampling from the underlying distribution. In particular, \((X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the distribution, and the corresponding sample mean is \[ M_n = \frac{Y_n}{n} = \frac{1}{n} \sum_{i=1}^n X_i \] By the law of large numbers, \(M_n \to \mu\) as \(n \to \infty\) with probability 1.
Stationary, Independent Increments
The partial sum process corresponding to a sequence of independent, identically distributed variables has two important properties, and these properties essentially characterize such processes.
If \(m \le n\) then \(Y_n - Y_m\) has the same distribution as \(Y_{n-m}\). Thus the process \(\bs{Y}\) has stationary increments .
Proof
Note that \(Y_n - Y_m = \sum_{i=m+1}^n X_i\) and is the sum of \(n - m\) independent variables, each with the common distribution. Of course, \(Y_{n-m}\) is also the sum of \(n - m\) independent variables, each with the common distribution.
Note however that \(Y_n - Y_m\) and \(Y_{n-m}\) are very different random variables; the theorem simply states that they have the same distribution .
If \(n_1 \le n_2 \le n_3 \le \cdots\) then \(\left(Y_{n_1}, Y_{n_2} - Y_{n_1}, Y_{n_3} - Y_{n_2}, \ldots\right)\) is a sequence of independent random variables. Thus the process \(\bs{Y}\) has independent increments .
Proof
The terms in the sequence of increments \(\left(Y_{n_1}, Y_{n_2} - Y_{n_1}, Y_{n_3} - Y_{n_2}, \ldots\right)\) are sums over disjoint collections of terms in the sequence \(\bs{X}\). Since the sequence \(\bs{X}\) is independent, so is the sequence of increments.
Conversely, suppose that \(\bs{V} = (V_0, V_1, V_2, \ldots)\) is a random process with stationary, independent increments. Define \(U_i = V_i - V_{i-1}\) for \(i \in \N_+\). Then \(\bs{U} = (U_1, U_2, \ldots)\) is a sequence of independent, identically distributed variables and \(\bs{V}\) is the partial sum process associated with \(\bs{U}\).
Thus, partial sum processes are the only discrete-time random processes that have stationary, independent increments. An interesting, and much harder problem, is to characterize the continuous-time processes that have stationary independent increments. The Poisson counting process has stationary independent increments, as does the Brownian motion process.
Moments
If \(n \in \N\) then
- \(\E(Y_n) = n \mu\)
- \(\var(Y_n) = n \sigma^2\)
Proof
The results follow from basic properties of expected value and variance. Expected value is a linear operation so \( \E(Y_n) = \sum_{i=1}^n \E(X_i) = n \mu \). By independence, \(\var(Y_n) = \sum_{i=1}^n \var(X_i) = n \sigma^2\).
If \(n \in \N_+\) and \(m \in \N\) with \(m \le n\) then
- \(\cov(Y_m, Y_n) = m \sigma^2\)
- \(\cor(Y_m, Y_n) = \sqrt{\frac{m}{n}}\)
- \(\E(Y_m Y_n) = m \sigma^2 + m n \mu^2\)
Proof
- Note that \(Y_n = Y_m + (Y_n - Y_m)\). This follows from basic properties of covariance, and Theorem 1 and Theorem 2 : \[ \cov(Y_m, Y_n) = \cov(Y_m, Y_m) + \cov(Y_m, Y_n - Y_m) = \var(Y_m) + 0 = m \sigma^2 \]
- This result follows from part (a) and Theorem 4 \[ \cor(Y_m, Y_m) = \frac{\cov(Y_m, Y_n)}{\sd(Y_m) \sd(Y_n)} = \frac{m \sigma^2}{\sqrt{m \sigma^2} \sqrt{n \sigma^2}} = \sqrt{\frac{m}{n}} \]
- This result also follows from part (a) and Theorem 4 : \(\E(Y_m Y_n) = \cov(Y_m, Y_n) + \E(Y_m) \E(Y_n) = m \sigma^2 + m \mu n \mu\)
If \(X\) has moment generating function \(G\) then \(Y_n\) has moment generating function \(G^n\).
Proof
This follows from a basic property of generating functions: the generating function of a sum of independent variables is the product of the generating functions of the terms.
Distributions
Suppose that \(X\) has either a discrete distribution or a continuous distribution with probability density function \(f\). Then the probability density function of \(Y_n\) is \(f^{*n} = f * f * \cdots * f\), the convolution power of \(f\) of order \(n\).
Proof
This follows from a basic property of PDFs: the pdf of a sum of independent variables is the convolution of the PDFs of the terms.
More generally, we can use the stationary and independence properties to find the joint distributions of the partial sum process:
If \(n_1 \lt n_2 \lt \cdots \lt n_k\) then \((Y_{n_1}, Y_{n_2}, \ldots, Y_{n_k})\) has joint probability density function \[ f_{n_1, n_2, \ldots, n_k}(y_1, y_2, \ldots, y_k) = f^{*n_1}(y_1) f^{*(n_2 - n_1)}(y_2 - y_1) \cdots f^{*(n_k - n_{k-1})}(y_k - y_{k-1}), \quad (y_1, y_2, \ldots, y_k) \in \R^k \]
Proof
This follows from the multivariate change of variables theorem.
The Central Limit Theorem
First, let's make the central limit theorem more precise. From Theorem 4 , we cannot expect \(Y_n\) itself to have a limiting distribution. Note that \(\var(Y_n) \to \infty\) as \(n \to \infty\) since \(\sigma \gt 0\), and \(\E(Y_n) \to \infty\) as \(n \to \infty\) if \(\mu \gt 0\) while \(\E(Y_n) \to -\infty\) as \(n \to \infty\) if \(\mu \lt 0\). Similarly, we know that \(M_n \to \mu\) as \(n \to \infty\) with probability 1, so the limiting distribution of the sample mean is degenerate. Thus, to obtain a limiting distribution of \(Y_n\) or \(M_n\) that is not degenerate, we need to consider, not these variables themeselves, but rather the common standard score . Thus, let \[ Z_n = \frac{Y_n - n \mu}{\sqrt{n} \sigma} = \frac{M_n - \mu}{\sigma \big/ \sqrt{n}} \]
\(Z_n\) has mean 0 and variance 1.
- \(\E(Z_n) = 0\)
- \(\var(Z_n) = 1\)
Proof
These results follow from basic properties of expected value and variance, and are true for the standard score associated with any random variable. Recall also that the standard score of a variable is invariant under linear transformations with positive slope. The fact that the standard score of \(Y_n\) and the standard score of \(M_n\) are the same is a special case of this.
The precise statement of the central limit theorem is that the distribution of the standard score \(Z_n\) converges to the standard normal distribution as \(n \to \infty\). Recall that the standard normal distribution has probability density function \[ \phi(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^2}, \quad z \in \R \] and is studied in more detail in the chapter on special distributions. A special case of the central limit theorem (to Bernoulli trials), dates to Abraham De Moivre. The term central limit theorem was coined by George Pólya in 1920. By definition of convergence in distribution, the central limit theorem states that \(F_n(z) \to \Phi(z)\) as \(n \to \infty\) for each \(z \in \R\), where \(F_n\) is the distribution function of \(Z_n\) and \(\Phi\) is the standard normal distribution function:
\[ \Phi(z) = \int_{-\infty}^z \phi(x) \, dx = \int_{-\infty}^z \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} \, dx, \quad z \in \R \]
An equivalent statment of the central limit theorm involves convergence of the corresponding characteristic functions. This is the version that we will give and prove, but first we need a generalization of a famous limit from calculus.
Suppose that \((a_1, a_2, \ldots)\) is a sequence of real numbers and that \(a_n \to a \in \R\) as \(n \to \infty\). Then \[ \left( 1 + \frac{a_n}{n} \right)^n \to e^a \text{ as } n \to \infty \]
Now let \(\chi\) denote the characteristic function of the standard score of the sample variable \(X\), and let \(\chi_n\) denote the characteristic function of the standard score \(Z_n\): \[ \chi(t) = \E \left[ \exp\left( i t \frac{X - \mu}{\sigma} \right) \right], \; \chi_n(t) = \E[\exp(i t Z_n)]; \quad t \in \R \] Recall that \(t \mapsto e^{-\frac{1}{2}t^2}\) is the characteristic function of the standard normal distribution.
The central limit theorem . The distribution of \(Z_n\) converges to the standard normal distribution as \(n \to \infty\). That is, \(\chi_n(t) \to e^{-\frac{1}{2}t^2}\) as \(n \to \infty\) for each \(t \in \R\).
Proof
Note that \(\chi(0) = 1\), \(\chi^\prime(0) = 0\), \(\chi^{\prime \prime}(0) = -1\). Next \[ Z_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n \frac{X_i - \mu}{\sigma} \] From properties of characteristic functions, \(\chi_n(t) = \chi^n (t / \sqrt{n})\) for \(t \in \R\). By Taylor's theorem (named after Brook Taylor), \[ \chi\left(\frac{t}{\sqrt{n}}\right) = 1 + \frac{1}{2} \chi^{\prime\prime}(s_n) \frac{t^2}{n} \text{ where } \left|s_n\right| \le \frac{\left|t\right|}{n} \] But \(s_n \to 0\) and hence \(\chi^{\prime\prime}(s_n) \to -1\) as \(n \to \infty\). Finally, \[ \chi_n(t) = \left[1 + \frac{1}{2} \chi^{\prime\prime}(s_n) \frac{t^2}{n} \right]^n \to e^{-\frac{1}{2} t^2} \text{ as } n \to \infty \]
Normal Approximations
The central limit theorem implies that if the sample size \(n\) is
large
then the distribution of the partial sum \(Y_n\) is approximately normal with mean \(n \mu\) and variance \(n \sigma^2\). Equivalently the sample mean \(M_n\) is approximately normal with mean \(\mu\) and variance \(\sigma^2 / n\). The central limit theorem is of fundamental importance, because it means that we can approximate the distribution of certain statistics, even if we know very little about the underlying sampling distribution.
Of course, the term
large
is relative. Roughly, the more
abnormal
the basic distribution, the larger \(n\) must be for normal approximations to work well. The rule of thumb is that a sample size \(n\) of at least 30 will usually suffice if the basic distribution is not too weird; although for many distributions smaller \(n\) will do.
Let \(Y\) denote the sum of the variables in a random sample of size 30 from the uniform distribution on \([0, 1]\). Find normal approximations to each of the following:
- \(\P(13 \lt Y \lt 18)\)
- The 90th percentile of \(Y\)
Answer
- 0.8682
- 17.03
Random variable \(Y\) in the previous exercise has the Irwin-Hall distribution of order 30. The Irwin-Hall distributions are studied in more detail in the chapter on Special Distributions and are named for Joseph Irwin and Phillip Hall.
In the special distribution simulator, select the Irwin-Hall distribution. Vary and \(n\) from 1 to 10 and note the shape of the probability density function. With \(n = 10\) run the experiment 1000 times and compare the empirical density function to the true probability density function.
Let \(M\) denote the sample mean of a random sample of size 50 from the distribution with probability density function \(f(x) = \frac{3}{x^4}\) for \(1 \le x \lt \infty\). This is a Pareto distribution , named for Vilfredo Pareto. Find normal approximations to each of the following:
- \(\P(M \gt 1.6)\)
- The 60th percentile of \(M\)
Answer
- 0.2071
- 1.531
The Continuity Correction
A slight technical problem arises when the sampling distribution is discrete. In this case, the partial sum also has a discrete distribution, and hence we are approximating a discrete distribution with a continuous one. Suppose that \(X\) takes integer values (the most common case) and hence so does the partial sum \(Y_n\). For any \(k \in \Z\) and \(h \in [0, 1)\), note that the event \(\{k - h \le Y_n \le k + h\}\) is equivalent to the event \(\{Y = k\}\). Different values of \(h\) lead to different normal approximations, even though the events are equivalent. The smallest approximation would be 0 when \(h = 0\), and the approximations increase as \(h\) increases. It is customary to split the difference by using \(h = \frac{1}{2}\) for the normal approximation. This is sometimes called the half-unit continuity correction or the histogram correction . The continuity correction is extended to other events in the natural way, using the additivity of probability.
Suppose that \(j, k \in \Z\) with \(j \le k\).
- For the event \(\{j \le Y_n \le k\} = \{j - 1 \lt Y_n \lt k + 1\}\), use \(\{j - \frac{1}{2} \le Y_n \le k + \frac{1}{2}\}\) in the normal approximation.
- For the event \(\{j \le Y_n\} = \{j - 1 \lt Y_n\}\), use \(\{j - \frac{1}{2} \le Y_n\}\) in the normal approximation.
- For the event \(\{Y_n \le k\} = \{Y_n \lt k + 1\}\), use \(\{Y_n \le k + \frac{1}{2}\}\) in the normal approximation.
Let \(Y\) denote the sum of the scores of 20 fair dice. Compute the normal approximation to \(\P(60 \le Y \le 75)\).
Answer
0.6741
In the dice experiment, set the die distribution to fair, select the sum random variable \(Y\), and set \(n = 20\). Run the simulation 1000 times and find each of the following. Compare with the result in the previous exercise:
- \(\P(60 \le Y \le 75)\)
- The relative frequency of the event \(\{60 \le Y \le 75\}\) (from the simulation)
Normal Approximation to the Gamma Distribution
Recall that the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) is a continuous distribution on \( (0, \infty) \) with probability density function \( f \) given by \[ f(x) = \frac{1}{\Gamma(k) b^k} x^{k-1} e^{-x/b}, \quad x \in (0, \infty) \] The mean is \( k b \) and the variance is \( k b ^2 \). The gamma distribution is widely used to model random times (particularly in the context of the Poisson model) and other positive random variables. The general gamma distribution is studied in more detail in the chapter on Special Distributions. In the context of the Poisson model (where \(k \in \N_+\)), the gamma distribution is also known as the Erlang distribution , named for Agner Erlang; it is studied in more detail in the chapter on the Poisson Process. Suppose now that \(Y_k\) has the gamma (Erlang) distribution with shape parameter \(k \in \N_+\) and scale parameter \(b \gt 0\) then \[ Y_k = \sum_{i=1}^k X_i \] where \((X_1, X_2, \ldots)\) is a sequence of independent variables, each having the exponential distribution with scale parameter \(b\). (The exponential distribution is a special case of the gamma distribution with shape parameter 1.) It follows that if \(k\) is large, the gamma distribution can be approximated by the normal distribution with mean \(k b\) and variance \(k b^2\). The same statement actually holds when \(k\) is not an integer. Here is the precise statement:
Suppose that \( Y_k \) has the gamma distribution with scale parameter \( b \in (0, \infty) \) and shape parameter \( k \in (0, \infty) \). Then the distribution of the standardized variable \( Z_k \) below converges to the standard normal distribution as \(k \to \infty\): \[ Z_k = \frac{Y_k - k b}{\sqrt{k} b} \]
In the special distribution simulator, select the gamma distribution. Vary and \(b\) and note the shape of the probability density function. With \(k = 10\) and various values of \(b\), run the experiment 1000 times and compare the empirical density function to the true probability density function.
Suppose that \(Y\) has the gamma distribution with shape parameter \(k = 10\) and scale parameter \(b = 2\). Find normal approximations to each of the following:
- \( \P(18 \le Y \le 23) \)
- The 80th percentile of \(Y\)
Answer
- 0.3063
- 25.32
Normal Approximation to the Chi-Square Distribution
Recall that the chi-square distribution with \(n \in (0, \infty)\) degrees of freedom is a special case of the gamma distribution, with shape parameter \(k = n / 2\) and scale parameter \(b = 2\). Thus, the chi-square distribution with \(n\) degrees of freedom has probability density function \[ f(x) = \frac{1}{\Gamma(n/2) 2^{n/2}} x^{n/2 - 1}e^{-x/2}, \quad 0 \lt x \lt \infty \] When \( n \) is a positive, integer, the chi-square distribution governs the sum of \( n \) independent, standard normal variables. For this reason, it is one of the most important distributions in statistics. The chi-square distribution is studied in more detail in the chapter on Special Distributions. From the previous discussion , it follows that if \(n\) is large, the chi-square distribution can be approximated by the normal distribution with mean \(n\) and variance \(2 n\). Here is the precise statement:
Suppose that \(Y_n\) has the chi-square distribution with \(n \in (0, \infty) \) degrees of freedom. Then the distribution of the standardized variable \( Z_n \) below converges to the standard normal distribution as \(n \to \infty\): \[ Z_n = \frac{Y_n - n}{\sqrt{2 n}} \]
In the special distribution simulator, select the chi-square distribution. Vary \(n\) and note the shape of the probability density function. With \(n = 20\), run the experiment 1000 times andcompare the empirical density function to the probability density function.
Suppose that \(Y\) has the chi-square distribution with \(n = 20\) degrees of freedom. Find normal approximations to each of the following:
- \(\P(18 \lt Y \lt 25)\)
- The 75th percentile of \(Y\)
Answer
- 0.4107
- 24.3
Normal Approximation to the Binomial Distribution
Recall that a Bernoulli trials sequence , named for Jacob Bernoulli, is a sequence \( (X_1, X_2, \ldots) \) of independent, identically distributed indicator variables with \(\P(X_i = 1) = p\) for each \(i\), where \(p \in (0, 1)\) is the parameter. In the usual language of reliability, \(X_i\) is the outcome of trial \(i\), where 1 means success and 0 means failure. The common mean is \(p\) and the common variance is \(p (1 - p)\).
Let \(Y_n = \sum_{i=1}^n X_i\), so that \(Y_n\) is the number of successes in the first \(n\) trials. Recall that \(Y_n\) has the binomial distribution with parameters \(n\) and \(p\), and has probability density function \[ f(k) = \binom{n}{k} p^k (1 - p)^{n-k}, \quad k \in \{0, 1, \ldots, n\} \] The binomial distribution is studied in more detail in the chapter on Bernoulli trials.
It follows from the central limit theorem that if \(n\) is large, the binomial distribution with parameters \(n\) and \(p\) can be approximated by the normal distribution with mean \(n p\) and variance \(n p (1 - p)\). The rule of thumb is that \(n\) should be large enough for \(n p \ge 5\) and \(n (1 - p) \ge 5\). (The first condition is the important one when \(p \lt \frac{1}{2}\) and the second condition is the important one when \(p \gt \frac{1}{2}\).) Here is the precise statement:
Suppose that \( Y_n \) has the binomial distribution with trial parameter \( n \in \N_+ \) and success parameter \( p \in (0, 1) \). Then the distribution of the standardized variable \(Z_n\) given below converges to the standard normal distribution as \(n \to \infty\): \[ Z_n = \frac{Y_n - n p}{\sqrt{n p (1 - p)}} \]
In the binomial timeline experiment, vary \(n\) and \(p\) and note the shape of the probability density function. With \(n = 50\) and \(p = 0.3\), run the simulation 1000 times and compute the following:
- \(\P(12 \le Y \le 16)\)
- The relative frequency of the event \(\{12 \le Y \le 16\}\) (from the simulation)
Answer
- 0.5448
Suppose that \(Y\) has the binomial distribution with parameters \(n = 50\) and \(p = 0.3\). Compute the normal approximation to \( \P(12 \le Y \le 16) \) (don't forget the continuity correction) and compare with the results of the previous exercise.
Answer
0.5383
Normal Approximation to the Poisson Distribution
Recall that the
Poisson distribution
, named for Simeon Poisson, is a discrete distribution on \( \N \) with probability density function \( f \) given by \[ f(x) = e^{-\theta} \frac{\theta^x}{x!}, \quad x \in \N \] where \(\theta \gt 0\) is a parameter. The parameter is both the mean and the variance of the distribution. The Poisson distribution is widely used to model the number of
random points
in a region of time or space, and is studied in more detail in the chapter on the Poisson Process. In this context, the parameter is proportional to the size of the region.
Suppose now that \(Y_n\) has the Poisson distribution with parameter \(n \in \N_+\). Then \[ Y_n = \sum_{i=1}^n X_i \] where \((X_1, X_2, \ldots, X_n)\) is a sequence of independent variables, each with the Poisson distribution with parameter 1. It follows from the central limit theorem that if \(n\) is large, the Poisson distribution with parameter \(n\) can be approximated by the normal distribution with mean \(n\) and variance \(n\). The same statement holds when the parameter \(n\) is not an integer. Here is the precise statement:
. Suppose that \( Y_\theta \) has the Poisson distribution with parameter \( \theta \in (0, \infty) \). Then the distribution of the standardized variable \( Z_\theta \) below converges to the standard normal distribution as \(\theta \to \infty\):
\[ Z_\theta = \frac{Y_\theta - \theta}{\sqrt{\theta}} \]Suppose that \(Y\) has the Poisson distribution with mean 20.
- Compute the true value of \(\P(16 \le Y \le 23)\).
- Compute the normal approximation to \(\P(16 \le Y \le 23)\).
Answer
- 0.6310
- 0.6259
In the Poisson experiment, vary the time and rate parameters \(t\) and \(r\) (the parameter of the Poisson distribution in the experiment is the product \(r t\)). Note the shape of the probability density function. With \(r = 5\) and \(t = 4\), run the experiment 1000 times and compare the empirical density function to the true probability density function.
Normal Approximation to the Negative Binomial Distribution
The general version of the negative binomial distribution is a discrete distribution on \( \N \), with shape parameter \( k \in (0, \infty) \) and success parameter \( p \in (0, 1) \). The probability density function \( f \) is given by \[ f(n) = \binom{n + k - 1}{n} p^k (1 - p)^n, \quad n \in \N_+ \] The mean is \( k (1 - p) / p \) and the variance is \( k (1 - p) / p^2 \). The negative binomial distribution is studied in more detail in the chapter on Bernoulli trials. If \( k \in \N_+ \), the distribution governs the number of failures \( Y_k \) before success number \( k \) in a sequence of Bernoulli trials with success parameter \( p \). Thus in this case, \[ Y_k = \sum_{i=1}^k X_i \] where \((X_1, X_2, \ldots, X_k)\) is a sequence of independent variables, each having the geometric distribution on \(\N\) with parameter \(p\). (The geometric distribution is a special case of the negative binomial, with parameters 1 and \(p\).) In the context of the Bernoulli trials, \( X_1 \) is the number of failures before the first success, and for \( i \in \{2, 3, \ldots\} \), \(X_i\) is the number of failures between success number \( i - 1 \) success number \( i \). It follows that if \(k\) is large, the negative binomial distribution can be approximated by the normal distribution. The same statement holds if \( k \) is not an integer. Here is the precise statement:
Suppose that \( Y_k \) has the negative binomial distribution with shape parameter \( k \in (0, 1) \) and scale parameter \( p \in (0, 1) \). Then the distribution of the standardized variable \( Z_k \) below converges to the standard normal distribution as \(k \to \infty\): \[ Z_k = \frac{p Y_k - k(1 - p)}{\sqrt{k (1 - p)}} \]
Another version of the negative binomial distribution is the distribution of the trial number \( V_k \) of success number \( k \in \N_+ \). So \( V_k = k + Y_k \) and \( V_k \) has mean \( k / p \) and variance \( k (1 - p) / p^2 \). The normal approximation applies to the distribution of \( V_k \) as well, if \( k \) is large, and since the distributions are related by a location transformation, the standard scores are the same. That is \[ \frac{p V_k - k}{\sqrt{k (1 - p)}} = \frac{p Y_k - k(1 - p)}{\sqrt{k ( 1 - p)}} \]
In the negative binomial experiment, vary \(k\) and \(p\) and note the shape of the probability density function. With \(k = 5\) and \(p = 0.4\), run the experiment 1000 times and compare the empirical density function to the true probability density function.
Suppose that \(Y\) has the negative binomial distribution with trial parameter \(k = 10\) and success parameter \(p = 0.4\). Find normal approximations to each of the following:
- \(\P(20 \lt Y \lt 30)\)
- The 80th percentile of \(Y\)
Answer
- 0.6318
- 30.1
Partial Sums with a Random Number of Terms
Our last topic is a bit more esoteric, but still fits with the general setting of this section. Recall that \(\bs{X} = (X_1, X_2, \ldots)\) is a sequence of independent, identically distributed real-valued random variables with common mean \(\mu\) and variance \(\sigma^2\). Suppose now that \(N\) is a random variable (on the same probability space) taking values in \(\N\), also with finite mean and variance. Then \[ Y_N = \sum_{i=1}^N X_i \] is a random sum of the independent, identically distributed variables. That is, the terms are random of course, but so also is the number of terms \(N\). We are primarily interested in the moments of \(Y_N\).
Independent Number of Terms
Suppose first that \(N\), the number of terms, is independent of \(\bs{X}\), the sequence of terms. Computing the moments of \(Y_N\) is a good exercise in conditional expectation.
The conditional expected value of \(Y_N\) given \(N\), and the expected value of \(Y_N\) are
- \(\E(Y_N \mid N) = N \mu\)
- \(\E(Y_N) = \E(N) \mu\)
The conditional variance of \(Y_N\) given \(N\) and the variance of \(Y_N\) are
- \(\var(Y_N \mid N) = N \sigma^2\)
- \(\var(Y_N) = \E(N) \sigma^2 + \var(N) \mu^2\)
Let \(H\) denote the probability generating function of \(N\). Show that the moment generating function of \(Y_N\) is \(H \circ G\).
- \(\E(e^{t Y_N} \mid N) = [G(t)]^N\)
- \(\E(e^{t Y_N}) = H(G(t))\)
Wald's Equation
The result in Exercise 29 (b) generalizes to the case where the random number of terms \(N\) is a stopping time for the sequence \(\bs{X}\). This means that the event \(\{N = n\}\) depends only on (technically, is measurable with respect to) \((X_1, X_2, \ldots, X_n)\) for each \(n \in \N\). The generalization is knowns as Wald's equation , and is named for Abraham Wald. Stopping times are studied in much more technical detail in the section on Filtrations and Stopping Times.
If \(N\) is a stopping time for \(\bs{X}\) then \(\E(Y_N) = \E(N) \mu\).
Proof
First note that \(Y_N = \sum_{i=1}^\infty X_i \bs{1}(i \le N)\). But \(\{i \le N\} = \{N \lt i\}^c\) depends only on \(\{X_1, \ldots, X_{i-1}\}\) and hence is independent of \(X_i\). Thus \(\E[X_i \bs{1}(i \le N)] = \mu \P(N \ge i)\). Suppose that \(X_i \ge 0\) for each \(i\). Taking expected values term by term gives Wald's equation in this special case. The interchange of sum and expected value is justified by the monotone convergence theorem. Now Wald's equation can be established in general by using the dominated convergence theorem.
An elgant proof of Wald's equation is given in the chapter on Martingales.
Suppose that the number of customers arriving at a store during a given day has the Poisson distribution with parameter 50. Each customer, independently of the others (and independently of the number of customers), spends an amount of money that is uniformly distributed on the interval \([0, 20]\). Find the mean and standard deviation of the amount of money that the store takes in during a day.
Answer
500, 81.65
When a certain critical component in a system fails, it is immediately replaced by a new, statistically identical component. The components are independent, and the lifetime of each (in hours) is exponentially distributed with scale parameter \(b\). During the life of the system, the number of critical components used has a geometric distribution on \(\N_+\) with parameter \(p\). For the total life of the critical component,
- Find the mean.
- Find the standard deviation.
- Find the moment generating function.
- Identify the distribution by name.
Answer
- \(b / p\)
- \(b / p\)
- \(t \mapsto \frac{1}{1 - (b/p)t}\)
- Exponential distribution with scale parameter \(b / p\)


12.2A: Immunoassays for Disease
- Describe how immunoassays aid in the diagnosis of disease
The Immune System
Immunology is the study of molecules, cells, and organs that make up the immune system. The function of the immune system is to recognize self antigens from non-self antigens and defend the body against non-self (foreign) agents. Through specific and non-specific defense mechanisms, the body’s immune system is able to react to microbial pathogens and protect against disease. The first line of defense against infection is intact skin, mucosal membrane surfaces, and secretions that prevent pathogens from penetrating into the body.
When a foreign agent penetrates the first line of resistance, an immune reaction is elicited and immune cells are recruited into the site of infection to clear microorganisms and damaged cells by phagocytosis. If the inflammation remains aggravated, antibody-mediated immune reaction is activated and different types of immune cells are engaged to resolve the disease. The immune system is composed of cellular and humoral elements. The cellular component includes mast cells, neutrophils, macrophages, T and B lymphocytes, and plasma cells. The humoral component includes complement, lyzozyme, interferon, antibodies, and cytokines. All work cooperatively to eliminate immunogenic foreign substances from the body.
Immunoassays
To aid in the diagnosis of disease caused by infectious microorganisms, immunoassays have been developed. These biochemical and serological techniques are based on the detection and quantitation of antibodies generated against an infectious agent, a microbe, or non-microbial antigen.
Because antibodies can be produced against any type of macromolecule, antibody-based techniques are useful in identifying molecules in solution or in cells. A blood sample is collected from the patient during the acute phase of the disease when antibody levels are high. Serum is then isolated and the concentration of antibodies is measured through various methods. Most assays rely on the formation of large immune complexes when an antibody binds to a specific antigen which can be detected in solution or in gels. Recent methods employ pure antibodies or antigens that have been immobilized on a platform and that can be measured using an indicator molecule. These methods provide high sensitivity and specificity and have become standard techniques in diagnostic immunology.
Key Points
- When microbial agents penetrate the body, they elicit an immune response that involves cellular and humoral components.
- An immune response is usually characterized by antibody secretions. These can be measured in the laboratory through various biochemical and serological techniques.
- Most immunoassays rely on the formation of antibody- antigen complexes that can be identified using an indicator molecule.
Key Terms
- humoral : Of or relating to the body fluids or humours.
- antibody : A protein produced by B-lymphocytes that binds to a specific antigen.


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Beatty, J. F., Samuelson, S. S., & Abril, P. S. (2018) . Business law and the legal environment. Cengage Learning.
Driesen, D. M. (2016) . The economic/noneconomic activity distinction under the commerce clause. Case W. Res. L. Rev., 67, 337.
United States v. Lopez , 514 U.S. 549, 558–559 (1995).
Beatty, J. F., Samuelson, S. S., & Abril, P. S. (2018) . Business law and the legal environment. Cengage Learning.
McAdams, T., Neslund, N., Zucker, K. D., & Neslund, K. (2015) . Law, business, and society. McGraw-Hill Education.
The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>. Download for free at https://openstax.org/details/books/b...w-i-essentials .


6.12B: Rate of Microbial Death
- Describe microbial death rates
The rate of microbial death can be determined. It is important in order to develop standard protocols for disinfection which will facilitate the sterilization routine in many industries. The goal is to find out what is the minimum time needed to achieve acceptable level of sterilization for a specific purpose. The killing agent can be different (e.g., heat, chemical with certain concentration) depending on the specific application.
When the killing factor is heat, the phrase thermal death can be used. Thermal death time is a concept used to determine how long it takes to kill a specific bacteria at a specific temperature. It was originally developed for food canning and has found applications in cosmetics, and in producing salmonella-free feeds for animals (e.g. poultry, and pharmaceuticals).
In the food industry, it is important to reduce the amount of microbes in products to ensure proper food safety. This is usually done by thermal processing and finding ways to reduce the number of bacteria in the product. Time-temperature measurements of bacterial reduction is determined by a D-value, meaning how long it would take to reduce the bacterial population by 90% or one log 10 at a given temperature. This D-value reference (D R ) point is 121°C.
Z or z-value is used to determine the time values with different D-values at different temperatures with its equation shown below:
\[z=T_2−T_1 \log D_1−\log D_2 z=T_2−T_1 \log D_1−\log D_2\]
where T is temperature in °C. Such death curves can be empirically established for all bactericidal agents. This D-value is affected by pH of the product where low pH has faster D values on various foods. The D-value at an unknown temperature can be calculated knowing the D-value at a given temperature provided the Z-value is known. The target of reduction in canning is the 12-D reduction of Clostridium botulinum , which means that processing time will reduce the amount of this bacteria by 10 12 bacteria per gram or milliliter. The D R for C. botulinum is 12.6 seconds. A 12-D reduction will take 151 seconds.
Key Points
- When researching microbial death rate, the goal is usually to find out the minimum time needed to achieve acceptable level of sterilization for a specific purpose.
- Bacterial reduction is determined by a D-value, meaning how long it would take to reduce the bacterial population by 90% or one log10 at a given state of the killing agent.
- Microbial death curves have been developed for many agents and are used in numerous industries.
Key Terms
- D-value : The time needed to reduce the bacterial population by 90% or one log10 at a given temperature.
- 12-D reduction : The time needed to reduce the amount of bacteria by 1012 bacteria per gram or milliliter.


16.13: Record-Keeping Requirements
- Summarize OSH Act record-keeping requirements
In general, employers with more than 10 employees are required to keep a record of serious work-related injuries and illnesses. A recordable injury or illness includes the following: [1]
- Any work-related fatality.
- Any work-related injury or illness that results in loss of consciousness, days away from work, restricted work, or transfer to another job.
- Any work-related injury or illness requiring medical treatment beyond first aid.
- Any work-related diagnosed case of cancer, chronic irreversible diseases, fractured or cracked bones or teeth, and punctured eardrums.
- There are special recording criteria for work-related cases involving needlesticks and sharps injuries ; medical removal ; hearing loss ; and tuberculosis ; refer to the linked page for specific criteria.
Minor injuries—specifically, those requiring first aid only—do not need to be recorded. Employers in low-risk industries are exempted from the recording rule unless otherwise notified in writing. Industries that are designated as low risk are identified by North American Industry Classification System (NAICS) code and listed on the Partially Exempt page on OSHA’s website. Examples of industries that are exempt from recording include Florists, Health and Personal Care Stores, Legal Services and Junior Colleges.
Note that “all employers, including those partially exempted by reason of company size or industry classification, must report to OSHA any workplace incident that results in a fatality, in-patient hospitalization, amputation, or loss of an eye.” [2]
Employers must report any worker fatality within 8 hours and any amputation, loss of an eye, or hospitalization of a worker within 24 hours. Reports can be submitted online or by calling the nearest OSHA office or OSHA’s 24-hour hotline.
Injury and illness records must be maintained on site for at least five years. Each February through April, employers are required to post a summary of the injuries and illnesses recorded the previous year. If requested, copies of the Log of Work-Related Injuries and Illnesses ( OSHA Form 300 ) must be provided to current and former employees, or their representatives. Employers must also provide employees or their authorized representatives access to their medical records and exposure records.
-
"
OSHA Injury and Illness Record-Keeping and Reporting Requirements
." Occupational Safety and Health Administration, United States Department of Labor. Accessed August 20, 2019. ↵
-
"
Non-Mandatory Appendix A to Subpart B – Partially Exempt Industries
." Occupational Safety and Health Administration, United States Department of Labor. Accessed August 20, 2019. ↵
Contributors and Attributions
- Beauty in Flowers. Authored by : Roman Kraft. Located at : https://unsplash.com/photos/_oH66az_yug . License : CC0: No Rights Reserved . License Terms : Unsplash License
- Ruby Loftus screwing a Breech-ring. Authored by : Laura Knight. Provided by : Wikimedia Commons. Located at : https://commons.wikimedia.org/wiki/File:Ruby_Loftus_screwing_a_Breech-ring_(1943)_(Art._IWM_LD_2850).jpg . License : Public Domain: No Known Copyright


7.1: Mediation and Moderation Models
At the end of this section you should be able to answer the following questions:
- Define the concept of a moderator variable.
- Define the concept of a mediator variable.
As we discussed in the lesson on correlations and regressions, understanding associations between psychological constructs can tell researchers a great deal about how certain mental health concerns and behaviours affects us on an emotional level. Correlation analyses focus on the relationship between two variables, and regression is the association of multiple independent variables with a single dependant variable.
Some predictor variables interact in a sequence, rather than impacting the outcome variable singly or as a group (like regression).
Moderation and mediation is a form of regression that allows researchers to analyse how a third variable effects the relationship of the predictor and outcome variable.
PowerPoint: Basic Mediation Model
Consider the Basic Mediation Model in this slide:
We know that high levels of stress can negatively impact health, we also know that a high level of social support can be beneficial to health. With these two points of knowledge, could it be that social support might provide a protective factor from the effects of stress on health? Thinking about a sequence of effects, perhaps social support can mediate the effect of stress on health.
Mediation is a more complicated extension of multiple regression procedures. Mediation examines the pattern of relationships among three variables (Simple Mediation Model), and can be used on four or more variables.
Examples of Research Questions
Here are some examples of research questions that could use a mediation analysis.
- If an intervention increases secure attachment among young children, do behavioural problems decrease when the children enter school?
- Does physical abuse in early childhood lead to deviant processing of social information that leads to aggressive behaviour?
- Do performance expectations start a self-fulfilling prophecy that affects behaviour?
- Can changes in cognitive attributions reduce depression?
PowerPoint: Three Mediation Figures
Consider the Three Figures Illustrating Mediation from the following slides:
Looking at this conceptual model, you can see the direct effect of X on Y. You can also see the effect of M on Y. What we are interested in is the effects of X on Y, accounting for the effects of M.
An example mediation model is that of the mediating effect of health-related behaviours on conscientiousness and overall physical health. Conscientiousness, or the personality trait associated with hardworking has relationship with overall physical health, but if an individual is hardworking, but does not perform health-related behaviours like exercise or diet control, then they are likely to be less healthy. From this, we can assume that health-related behaviours mediates the relationship between conscientiousness and physical health.


27.3: Phospholipids
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- Steven Farmer, Dietmar Kennepohl, & William Reusch
- University of Illinois Springfield
Learning Objectives
After completing this section, you should be able to
- draw the general structure of a phosphoglyceride.
- describe the occurrence and importance of phosphoglycerides in plant and animal tissues.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- phosphoglyceride
- phospholipid
Phosphate Esters
Similarly to carboxylic acids, inorganic acids such as nitric acid (HNO 3 ), sulfuric acid (H 2 SO 4 ), and phosphoric acid (H 3 PO 4 ) also can form esters. The esters of phosphoric acid are especially important in biochemistry. A phosphoric acid molecule can form a monoalkyl, a dialkyl, or a trialkyl ester by reaction with one, two, or three molecules of an alcohol.
Esters of pyrophosphoric acid and triphosphoric acid are also important in biochemistry.
Phospholipids
The two main types of phospholipids are glycerophospholipids and sphingomyelins . Glycerophospholipids are important components of the lipid bilayer of cellular membranes. Glycerophospholipids are structurally related to fats, as both are derived from phosphatidic acid, which has the basic structure of glycerol with two ester bonds with fatty acids and one ester bond with phosphoric acid.
Phosphatidic acid is converted into a glycerophospholipid by esterifying various groups, such as ethanolamine, serine, choline, inositol, and others to the phosphate of phosphatidic acid. These compounds have a chiral center at C2 and have an R configuration when found in nature.
The phosphatidylethanolamines are found in all living cells and are one of the most common phosphatides. They are common constituents of brain tissue and in the spinal cord, making up as much as 45% of the total phospholipids. Phosphatidylethanolamines are asymmetrically distributed across membranes, being preferentially located on the inner leaflet (closest to the cytoplasm) of the plasma membrane.
Metabolically, phosphatidylethanloamines are precursors of phosphatidylcholines . Phosphatidylcholines are another group of important membrane components. They tend to be found more commonly on the outer leaflet of the plasma membrane. Nutritionally, the compounds are readily obtained from eggs and soybeans. Phosphatidylcholines are moved across membranes by Phosphatidylcholine transfer protein (PCTP).
Like the phosphatidylethanolamines, phosphatidylserines are preferentially located on the inner leaflet of the plasma membrane. When apoptosis (cell suicide) occurs, the preferential distribution is lost and the phosphatidylserines appear on the outer leaflet where they serve as a signal to macrophages to bind and destroy the cell.
Phosphatidylserine, a polar phospholipid. A serine head group has been added on to a phosphate group on a glycerol functional group, all shown in orange. The fatty acid chains are in blue.
Sphingomyelins, the simplest sphingolipids, each contain a fatty acid, a phosphoric acid, sphingosine, and choline. Because they contain phosphoric acid, they are also classified as phospholipids. Sphingomyelins are important constituents of the myelin sheath surrounding the axon of a nerve cell. Multiple sclerosis is one of several diseases resulting from damage to the myelin sheath.
Lipid Bilayer 20.
In one sense the polar lipids are like the anions of fatty acids, only more so. They contain two hydrophobic hydrocarbon tails and a head which may have several electrically charged sites. As in the case of soap and detergent molecules, the tails of polar lipids tend to avoid water and other polar substances, but the heads are quite compatible with such environments.
Lipid bilayer closeup Image by Aleia Kim
As ionic amphiphiles, phospholipids aggregate or self-assemble when mixed with water, but in a different manner than the soaps and detergents. Because of the two pendant alkyl chains present in phospholipids and the unusual mixed charges in their head groups, micelle formation is unfavorable relative to a bilayer structure. If a phospholipid is smeared over a small hole in a thin piece of plastic immersed in water, a stable planar bilayer of phospholipid molecules is created at the hole. As shown in the following diagram, the polar head groups on the faces of the bilayer contact water, and the hydrophobic alkyl chains form a nonpolar interior.
Organization of the lipid bilayer Image by Aleia Ki
The polar lipids are most commonly found as components of cell walls and other membranes. Nearly all hypotheses regarding membrane structure take as a fundamental component a lipid bilayer. Bilayers made in the laboratory have many properties in common with membranes. Ions such as Na + , K + , and Cl – cannot pass through them, but water molecules can. Certain carrier molecules can transport K + and other ions across a bilayer, apparently by wrapping a hydrophobic cloak around them to disguise their charges. Membrane proteins in a bilayer also allow for transport of ions and other molecules across the bilayer which could not cross otherwise.


6.1.5: Paley’s Watchmaker Argument
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17 Paley’s Watchmaker Argument
The Teleological Argument 26
CHAPTER I.
STATE OF THE ARGUMENT.
IN crossing a health, suppose I pitched my foot against a stone, and were asked how the stone came to be there; I might possibly answer, that, for any thing I knew to the contrary, it had lain there for ever: nor would it perhaps be very easy to show the absurdity of this answer. But suppose I had found a watch upon the ground, and it should be inquired how the watch happened to be in that place; I should hardly think of the answer which I had before given, that, for any thing I knew, the watch might have always been there. Yet why should not this answer serve for the watch as well as for the stone? why is it not as admissible in the second case, as in the first? For this reason, and for no other, viz. that, when we come to inspect the watch, we perceive (what we could not discover in the stone) that its several parts are framed and put together for a purpose, e. g. that they are so formed and adjusted as to produce motion, and that motion so regulated as to point out the hour of the day; that, if the different parts had been differently shaped from what they are, of a different size from what they are, or placed after any other manner, or in any other order, than that in which they are placed, either no motion at all would have been carried on in the machine, or none which would have answered the use that is now served by it. To reckon up a few of the plainest of these parts, and of their offices, all tending to one result:-- We see a cylindrical box containing a coiled elastic spring, which, by its endeavour to relax itself, turns round the box. We next observe a flexible chain (artificially wrought for the sake of flexure), communicating the action of the spring from the box to the fusee. We then find a series of wheels, the teeth of which catch in, and apply to, each other, conducting the motion from the fusee to the balance, and from the balance to the pointer; and at the same time, by the size and shape of those wheels, so regulating that motion, as to terminate in causing an index, by an equable and measured progression, to pass over a given space in a given time. We take notice that the wheels are made of brass in order to keep them from rust; the springs of steel, no other metal being so elastic; that over the face of the watch there is placed a glass, a material employed in no other part of the work, but in the room of which, if there had been any other than a transparent substance, the hour could not be seen without opening the case. This mechanism being observed (it requires indeed an examination of the instrument, and perhaps some previous knowledge of the subject, to perceive and understand it; but being once, as we have said, observed and understood), the inference, we think, is inevitable, that the watch must have had a maker: that there must have existed, at some time, and at some place or other, an artificer or artificers who formed it for the purpose which we find it actually to answer; who comprehended its construction, and designed its use.
I. Nor would it, I apprehend, weaken the conclusion, that we had never seen a watch made; that we had never known an artist capable of making one; that we were altogether incapable of executing such a piece of workmanship ourselves, or of understanding in what manner it was performed; all this being no more than what is true of some exquisite remains of ancient art, of some lost arts, and, to the generality of mankind, of the more curious productions of modern manufacture. Does one man in a million know how oval frames are turned? Ignorance of this kind exalts our opinion of the unseen and unknown artist's skill, if he be unseen and unknown, but raises no doubt in our minds of the existence and agency of such an artist, at some former time, and in some place or other. Nor can I perceive that it varies at all the inference, whether the question arise concerning a human agent, or concerning an agent of a different species, or an agent possessing, in some respects, a different nature.
II. Neither, secondly, would it invalidate our conclusion, that the watch sometimes went wrong, or that it seldom went exactly right. The purpose of the machinery, the design, and the designer, might be evident, and in the case supposed would be evident, in whatever way we accounted for the irregularity of the movement, or whether we could account for it or not. It is not necessary that a machine be perfect, in order to show with what design it was made: still less necessary, where the only question is, whether it were made with any design at all.
III. Nor, thirdly, would it bring any uncertainty into the argument, if there were a few parts of the watch, concerning which we could not discover, or had not yet discovered, in what manner they conduced to the general effect; or even some parts, concerning which we could not ascertain, whether they conduced to that effect in any manner whatever. For, as to the first branch of the case; if by the loss, or disorder, or decay of the parts in question, the movement of the watch were found in fact to be stopped, or disturbed, or retarded, no doubt would remain in our minds as to the utility or intention of these parts, although we should be unable to investigate the manner according to which, or the connexion by which, the ultimate effect depended upon their action or assistance; and the more complex is the machine, the more likely is this obscurity to arise. Then, as to the second thing supposed, namely, that there were parts which might be spared, without prejudice to the movement of the watch, and that we had proved this by experiment,--these superfluous parts, even if we were completely assured that they were such, would not vacate the reasoning which we had instituted concerning other parts. The indication of contrivance remained, with respect to them, nearly as it was before.
IV. Nor, fourthly, would any man in his senses think the existence of the watch, with its various machinery, accounted for, by being told that it was one out of possible combinations of material forms; that whatever he had found in the place where he found the watch, must have contained some internal configuration or other; and that this configuration might be the structure now exhibited, viz. of the works of a watch, as well as a different structure.
V. Nor, fifthly, would it yield his inquiry more satisfaction to be answered, that there existed in things a principle of order, which had disposed the parts of the watch into their present form and situation. He never knew a watch made by the principle of order; nor can he even form to himself an idea of what is meant by a principle of order, distinct from the intelligence of the watch-maker.
VI. Sixthly, he would be surprised to hear that the mechanism of the watch was no proof of contrivance, only a motive to induce the mind to think so:
VII. And not less surprised to be informed, that the watch in his hand was nothing more than the result of the laws of metallicnature. It is a perversion of language to assign any law, as the efficient, operative cause of any thing. A law presupposes an agent; for it is only the mode, according to which an agent proceeds: it implies a power; for it is the order, according to which that power acts. Without this agent, without this power, which are both distinct from itself, the law does nothing; is nothing. The expression, "the law of metallic nature," may sound strange and harsh to a philosophic ear; but it seems quite as justifiable as some others which are more familiar to him, such as "the law of vegetable nature," "the law of animal nature," or indeed as "the law of nature" in general, when assigned as the cause of phænomena, in exclusion of agency and power; or when it is substituted into the place of these.
VIII. Neither, lastly, would our observer be driven out of his conclusion, or from his confidence in its truth, by being told that he knew nothing at all about the matter. He knows enough for his argument: he knows the utility of the end: he knows the subserviency and adaptation of the means to the end. These points being known, his ignorance of other points, his doubts concerning other points, affect not the certainty of his reasoning. The consciousness of knowing little, need not beget a distrust of that which he does know.


10.2E: Exercises
Practice Makes Perfect
In the following exercises, find
- \((f \circ g)(x)\)
- \((g \circ f)(x)\)
- \((f \cdot g)(x)\)
- \(f(x)=4 x+3\) and \(g(x)=2 x+5\)
- \(f(x)=3 x-1\) and \(g(x)=5 x-3\)
- \(f(x)=6 x-5\) and \(g(x)=4 x+1\)
- \(f(x)=2 x+7\) and \(g(x)=3 x-4\)
- \(f(x)=3 x\) and \(g(x)=2 x^{2}-3 x\)
- \(f(x)=2 x\) and \(g(x)=3 x^{2}-1\)
- \(f(x)=2 x-1\) and \(g(x)=x^{2}+2\)
- \(f(x)=4 x+3\) and \(g(x)=x^{2}-4\)
- Answer
-
1.
- \(8x+23\)
- \(8x+11\)
- \(8 x^{2}+26 x+15\)
3.
- \(24x+1\)
- \(24x-19\)
- \(24x^{2}+19x-5\)
5.
- \(6 x^{2}-9 x\)
- \(18 x^{2}-9 x\)
- \(6 x^{3}-9 x^{2}\)
7.
- \(2 x^{2}+3\)
- \(4 x^{2}-4 x+3\)
- \(2 x^{3}-x^{2}+4 x-2\)
In the following exercises, find the values described.
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For functions \(f(x)=2 x^{2}+3\) and \(g(x)=5x-1\), find
- \((f \circ g)(-2)\)
- \((g \circ f)(-3)\)
- \((f \circ f)(-1)\)
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For functions \(f(x)=5 x^{2}-1\) and \(g(x)=4x−1\), find
- \((f \circ g)(1)\)
- \((g \circ f)(-1)\)
- \((f \circ f)(2)\)
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For functions \(f(x)=2x^{3}\) and \(g(x)=3x^{2}+2\), find
- \((f \circ g)(-1)\)
- \((g \circ f)(1)\)
- \((g \circ g)(1)\)
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For functions \(f(x)=3 x^{3}+1\) and \(g(x)=2 x^{2}=3\), find
- \((f \circ g)(-2)\)
- \((g \circ f)(-1)\)
- \((g \circ g)(1)\)
- Answer
-
1.
- \(245\)
- \(104\)
- \(53\)
3.
- \(250\)
- \(14\)
- \(77\)
In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.
- \(\begin{array}{l}{\{(-3,9),(-2,4),(-1,1),(0,0)}, {(1,1),(2,4),(3,9) \}}\end{array}\)
- \(\begin{array}{l}{\{(9,-3),(4,-2),(1,-1),(0,0)}, {(1,1),(4,2),(9,3) \}}\end{array}\)
- \(\begin{array}{l}{\{(-3,-5),(-2,-3),(-1,-1)}, {(0,1),(1,3),(2,5),(3,7) \}}\end{array}\)
- \(\begin{array}{l}{\{(5,3),(4,2),(3,1),(2,0)}, {(1,-1),(0,-2),(-1,-3) \}}\end{array}\)
- Answer
-
1. Function; not one-to-one
3. One-to-one function
In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.
1.
-
Figure 10.1.65 -
Figure 10.1.66
2.
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Figure 10.1.67 -
Figure 10.1.68
3.
-
Figure 10.1.69 -
Figure 10.1.70
4.
-
Figure 10.1.71 -
Figure 10.1.72
- Answer
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1.
- Not a function
- Function; not one-to-one
3.
- One-to-one function
- Function; not one-to-one
In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.
- \(\{(2,1),(4,2),(6,3),(8,4)\}\)
- \(\{(6,2),(9,5),(12,8),(15,11)\}\)
- \(\{(0,-2),(1,3),(2,7),(3,12)\}\)
- \(\{(0,0),(1,1),(2,4),(3,9)\}\)
- \(\{(-2,-3),(-1,-1),(0,1),(1,3)\}\)
- \(\{(5,3),(4,2),(3,1),(2,0)\}\)
- Answer
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1. \(\begin{array}{l}{\text { Inverse function: }\{(1,2),(2,4),(3,6),(4,8)\} . \text { Domain: }\{1,2,3,4\} . \text { Range: }} {\{2,4,6,8\} .}\end{array}\)
3. \(\begin{array}{l}{\text { Inverse function: }\{(-2,0),(3,1),(7,2),(12,3)\} . \text { Domain: }\{-2,3,7,12\} \text { . }} {\text { Range: }\{0,1,2,3\}}\end{array}\)
5. \(\begin{array}{l}{\text { Inverse function: }\{(-3,-2),(-1,-1),(1,0),(3,1)\} . \text { Domain: }} {\{-3,-1,1,3\} . \text { Range: }\{-2,-1,0,1\}}\end{array}\)
In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.
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Figure 10.1.73 -
Figure 10.1.74 -
Figure 10.1.75 -
Figure 10.1.76
- Answer
-
1.
3.
In the following exercises, determine whether or not the given functions are inverses.
- \(f(x)=x+8\) and \(g(x)=x-8\)
- \(f(x)=x-9\) and \(g(x)=x+9\)
- \(f(x)=7 x\) and \(g(x)=\frac{x}{7}\)
- \(f(x)=\frac{x}{11}\) and \(g(x)=11 x\)
- \(f(x)=7 x+3\) and \(g(x)=\frac{x-3}{7}\)
- \(f(x)=5 x-4\) and \(g(x)=\frac{x-4}{5}\)
- \(f(x)=\sqrt{x+2}\) and \(g(x)=x^{2}-2\)
- \(f(x)=\sqrt[3]{x-4}\) and \(g(x)=x^{3}+4\)
- Answer
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1. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.
3. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.
5. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.
7. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses (for nonnegative \(x )\)
In the following exercises, find the inverse of each function.
- \(f(x)=x-12\)
- \(f(x)=x+17\)
- \(f(x)=9 x\)
- \(f(x)=8 x\)
- \(f(x)=\frac{x}{6}\)
- \(f(x)=\frac{x}{4}\)
- \(f(x)=6 x-7\)
- \(f(x)=7 x-1\)
- \(f(x)=-2 x+5\)
- \(f(x)=-5 x-4\)
- \(f(x)=x^{2}+6, x \geq 0\)
- \(f(x)=x^{2}-9, x \geq 0\)
- \(f(x)=x^{3}-4\)
- \(f(x)=x^{3}+6\)
- \(f(x)=\frac{1}{x+2}\)
- \(f(x)=\frac{1}{x-6}\)
- \(f(x)=\sqrt{x-2}, x \geq 2\)
- \(f(x)=\sqrt{x+8}, x \geq-8\)
- \(f(x)=\sqrt[3]{x-3}\)
- \(f(x)=\sqrt[3]{x+5}\)
- \(f(x)=\sqrt[4]{9 x-5}, x \geq \frac{5}{9}\)
- \(f(x)=\sqrt[4]{8 x-3}, x \geq \frac{3}{8}\)
- \(f(x)=\sqrt[5]{-3 x+5}\)
- \(f(x)=\sqrt[5]{-4 x-3}\)
- Answer
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1. \(f^{-1}(x)=x+12\)
3. \(f^{-1}(x)=\frac{x}{9}\)
5. \(f^{-1}(x)=6 x\)
7. \(f^{-1}(x)=\frac{x+7}{6}\)
9. \(f^{-1}(x)=\frac{x-5}{-2}\)
11. \(f^{-1}(x)=\sqrt{x-6}\)
13. \(f^{-1}(x)=\sqrt[3]{x+4}\)
15. \(f^{-1}(x)=\frac{1}{x}-2\)
17. \(f^{-1}(x)=x^{2}+2, x \geq 0\)
19. \(f^{-1}(x)=x^{3}+3\)
21. \(f^{-1}(x)=\frac{x^{4}+5}{9}, x \geq 0\)
23. \(f^{-1}(x)=\frac{x^{5}-5}{-3}\)
- Explain how the graph of the inverse of a function is related to the graph of the function.
- Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.
- Answer
-
1. Answers will vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


8.17: Reading- Earthquake Hazards
Earthquakes can be hazardous to humans and property in a variety of ways. Earthquake hazards arise from a combination of factors such as the size of the earthquake, distance to the epicenter, the underlying material and geologic structures, and building construction.
Potential Hazards
Ground Shaking
Ground shaking is caused by seismic waves During a significant earthquake, a particular location, and any building at that location, will be shaken by body waves (P- and S-) and surface waves (Rayleigh and Love. Each type of wave will have a different frequency (different number of waves passing by each second), and can shear or move a building in a different way, sometimes simultaneously. Adjacent neighborhoods or towns may experience very different intensities from the same earthquake, based on how far they are from the epicenter and what sort of rocks or sediments are in the ground beneath each area. Places underlain by thick deposits of unconsolidated sediments will experience a higher amplitude of shaking, at the same distance from the same earthquake, than places underlain by solid bedrock all the way to the surface. If unconsolidated sediments are fine-grained and wet, they may undergo liquefaction, increasing the damage to buildings and infrastructure and therefore increasing the intensity of the earthquake there. If unconsolidated sediments are overlain by a layer of artificial fill, the area is likely to experience more intense shaking and undergo more ground settling, and liquefaction if wet, than places that have not had a layer of artificial fill added.
Mexico City, one of the most populated cities in the world, is in a basin in the mountains of Mexico. Much of the city is built on artificial fill on top of fine-grained sediments from an extensive lake and wetlands that were drained and filled in as the city grew. As a result of how seismic waves are amplified in soft sediments, the shaking of the ground in Mexico City during an earthquake is greater than it is in areas outside the basin, which have bedrock close to the surface. During the 1985 earthquake, which originated offshore of Acapulco on the Pacific coast, 300 km (200 miles) away, many buildings in Mexico City collapsed and more than 20,000 people died.
The Marina District in northern San Francisco is built on artificial fill on wet bayshore sediments. Rubble and debris from buildings that collapsed or burned in that district as a result of the 1906 earthquake was used as artificial fill beneath structures built during the reconstruction of that area. In 1989, the Loma Prieta earthquake caused collapse of several buildings in the Marina District, and several people died there. Another example of a structure on wet ground covered by artificial fill that collapsed during the Loma Prieta earthquake is the Cypress viaduct in Oakland, CA. The USGS photo below shows the failed support columns.
Permanent Ground Displacement
During large earthquakes, the ground may permanently shift to a new position up, down, or sideways (up to 10 or more m, 30 or 40 ft, in extreme cases). This change in the location of the ground, which also tilts the ground, may cause disruption of roads and utilities and, in coastal cities, submergence or emergence of harbor facilities. Even ground shifts of less than a meter (a foot or two) can cause serious disruption to infrastructure.
Earth Rupture
During most earthquakes, some rupturing of the earth’s surface takes place along the fault trace. This produces a fault scarp, which may have up to several m (up to 10 ft or more) of vertical displacement. This can disrupt roads and utilities, and any buildings on a fault that ruptures may undergo extensive damage.
Rupture of the earth during an earthquake may also occur on secondary faults. Earthquake-induced rupture of the earth’s surface may also take place in weak zones of surface sediment that fracture and spread. If enough spreading of a ruptured surface layer takes place, it can be classified as a landslide. The NOAA photo below shows substantial damage in the Turnagain-By-The-Sea subdivision caused by the 1964 Great Alaskan Earthquake.
Landslides and Avalanches
On steep slopes and in mountainous areas, large earthquakes can set off many landslides, rockfalls, or avalanches. These can damage buildings, towns, or roads in the path of the landslides.
Liquefaction
If fine- or medium-grained, unconsolidated sediments are saturated with groundwater, the shaking that occurs during an earthquake may cause the sediment grains to lose contact with each other and become suspended in the water, temporarily turning what was solid ground into liquid ground. Building and other structures may sink, tilt, or slide a short distant in liquefied ground, causing serious damage.
Fires
Fires are a secondary rather than a primary effect of earthquakes. Broken electrical wires and natural gas pipes commonly set off fires during earthquakes. To compound the problem, the water supply may also be disrupted by earthquake damage, making it impossible to put the fire out with water from fire hydrants. The fire that broke out as a result of the great San Francisco earthquake of 1906 burned much of the city to the ground, causing more extensive damage to buildings than the shaking of the ground did during the earthquake.
Tsunamis
A tsunami is a set of waves in the ocean (or a large lake) with an extremely long wavelength, typically over 100 km long. Tsunamis move at many 100s of km per hour in deep water. Tsunamis can be set off by violent volcanic eruptions originating just below sea level, by giant landslides that either occur underwater or tumble into the sea from coastal mountains, by large meteorite impacts, and, most commonly, by earthquakes that greatly shake the ocean floor, which commonly happens at subduction zones.
The amplitude, or crest height, of an individual tsunami wave may be only about 1 m (roughly 3 feet high) in the open ocean. It is common for a tsunami to pass ships at sea without being noticed. However, as the wave approaches shore where the bottom grows shallower, the crest builds up to a height of up to several tens of meters (over 30 feet in some cases). The wave crest may wash ashore for several minutes before subsiding. Even a tsunami wave of no higher than 3 m (10 feet) coming ashore can cause extensive damage in harbors and to shores, as the long-wavelength wave keeps pouring in for several minutes.
Tsunamis consist of more than one wave, so a second wave crest may climb ashore several minutes later. Some tsunamis lead with the wave trough, so the first thing noticed as that type of tsunami approaches a shore is a dramatic drawback, or retreat, of the sea, like the tide suddenly going out. Such a drawback will inevitably be followed by a rising tsunami wave crest.
The tsunamis from the Sumatra earthquake of 2004 killed over 100,000 people in coastal areas of the Indian Ocean, some on shores several thousand km (several thousand miles) away from the epicenter of the earthquake. The great southeast Alaska earthquake of 1964 generated a tsunami that killed 16 people on the coast of Oregon and northern California, over 1,000 km (600 miles) away. Tsunamis have been known to cross the entire Pacific Ocean and cause fatalities a third of the way around the world. Hawaii, in the middle of the Pacific Ocean, has been damaged by tsunamis, originating from subduction earthquakes on the Pacific Rim, several times in the last few centuries. Japan, with its complex set of subduction zones and its eastern shoreline open to the Pacific Ocean, has experienced over 100 tsunamis in its recorded history, most recently as a result of a great subduction earthquake off the northern island in 2011 which resulted in several coastal towns being destroyed and thousands of people dying. The last great tsunami generated by the Cascadia subduction zone along the coast of the Pacific Northwest flooded coastal areas of Washington, Oregon northern California, and Vancouver Island in Canada. It occurred in 1701, long before the region was as populated as it is now. Subduction continues in the Cascadia subduction zone and more tsunamis can be expected to be generated by earthquakes there in the future.
Mitigating Earthquake Damage
Much can be done to reduce the risk of fatalities during earthquakes, and to reduce the damage to buildings and infrastructure; in other words, to mitigate the effects of earthquakes.
In many cases, it is collapsed buildings that cause the most harm during an earthquake. Buildings should be constructed in ways that make them unlikely to collapse during an earthquake. The strategies that engineers have developed include having sufficient flexibility in the structure to absorb shaking during an earthquake. Bricks, mortar, and concrete are rigid and brittle. However, bricks and mortar, and concrete, can be reinforced with steel to make them better able to survive an earthquake. Wood and steel are more flexible than bricks, mortar, and concrete, and lend themselves to the type of building that, properly designed and built according to code, is likely to survive an earthquake without collapsing.
The way a building is attached to its foundation, and how the foundation is anchored in the earth, are important considerations in earthquake design. Many houses built in the the early and mid-1900s in California were not attached to their foundations, based on the assumption that the weight of a house would keep it on its foundation. It turned out to be a bad assumption. Earthquakes caused houses to slide off their foundations. Many home-owners in the state have taken steps to make sure that their houses are now attached to their foundations; if a home-owner buys earthquake insurance, it is usually required by the insurance company that they do so. A large building or skyscraper built in an earthquake-prone area will normally have a great deal of flexibility built into it, including some sort of elastic strain absorption mechanism focused on points where the building attaches to its foundation.
Infrastructure—roads, bridges, utilities—can be built with margins of safety for the event of an earthquake. This includes gas pipelines designed to slide back and forth on their supports and having built-in shutoff valves that may be activated by automatic sensors, electric lines and grids with similar flexibility and shut-off capabilities, and roads, overpasses, and bridges built to withstand shaking during an earthquake.
Development and enforcement of building codes aimed at reducing risk from earthquakes often requires resources that are not available in impoverished regions. This leads to a higher likelihood that buildings will collapse from the same size earthquake in some areas of the world than in other areas.
Contributors and Attributions
- Basics -- Earthquakes. Authored by : Ralph L. Dawes and Cheryl D. Dawes. Provided by : Wenatchee Valley College. Located at : http://commons.wvc.edu/rdawes/G101OCL/Basics/earthquakes.html . Project : Geology 101 - Introduction to Physical Geology. License : CC BY: Attribution


9.14: Counseling, Psychological, and Social Services
School counselors, psychologists, and social workers can play a pivotal role in addressing the needs of individual students and their families, as well as supporting the social-emotional climate of the school as a whole. They can help identify areas of improvement for the school, and support SEL with specific lessons or elective classes, or by guiding student clubs. Family engagement can be included with workshops outside of school hours (CDC Healthy Schools, 2021). It is becoming increasingly apparent that adolescents can benefit from stress management skill development. Interventions that use yoga, mindfulness, and emotional regulation techniques can help students manage stress, and improve their mental health as well as academic performance. And employees can benefit from yoga and mindfulness interventions as well (Hawkins et al., 2023).


9.9: Why It Matters- Culture and Diversity
Why does a manager need to understand organizational culture and employee diversity?
If you get the culture right, most of the other stuff will just take care of itself
—Tony Hsieh, CEO of Zappos.com
Imagine for a moment that you have just been promoted to CEO of Zappos.com, a major online shoe and clothing shop. Where should you immediately focus your attention? Finance? Marketing? Maybe improving supply chain operations? Although all of these areas are important, Tony Hsieh, the current CEO of Zappos.com, believes that defining and maintaining the organizational culture is key. In fact, as you read in the earlier quote, he believes that the right culture will enable the other functional areas of business to succeed.
Why is culture so important? Think for a moment about teams that you have been a part of in the past. These could be teams at work, school, or even sports. Also think about how you felt as a member of the team. Now consider two questions: Did you share the same goals, desires, and values as the other members of the team? Did you feel like an important part of the team, and were you committed to its success?
There is a high probability that you had the same answer to both questions, whether yes or no. For example, if you shared the same values as others on the team, then you likely were highly committed to the group’s goals.
This is the strategic advantage of having a culture that resonates throughout the entire organization. It leads to a highly motivated team that is success-focused. It’s so important that Zappos.com lists on its corporate web page the Zappos 10 Core Values. It includes the following:
- Deliver WOW Through Service
- Embrace and Drive Change
- Create Fun and A Little Weirdness
- Be Adventurous, Creative, and Open-Minded
- Pursue Growth and Learning
- Build Open and Honest Relationships With Communication
- Build a Positive Team and Family Spirit
- Do More With Less
- Be Passionate and Determined
- Be Humble
By clearly defining its culture, Zappos.com is able to recruit the right people and have them focus on the right things. As you work through this module, you will gain a better understanding of the importance of an organization’s culture as a competitive advantage. Additionally, you will understand how to maintain culture over time or how implement cultural change if needed. Finally, you will come to see both the opportunities and challenges faced by promoting diversity within the culture.
Contributors and Attributions
- Why It Matters: Culture and Diversity. Authored by : Jeff Heflin and Lumen Learning. License : CC BY: Attribution
- Tony Hsieh. Authored by : Silicon Prairie News. Located at : https://www.flickr.com/photos/siliconprairienews/4637027240 . License : CC BY: Attribution


3.5: Chapter 3 Excercise
(5 pts)
What is best way to measure flow on a river every hour?
(5 pts)
What is the best way to estimate stormwater run-off from:
a. A building?
b. A neighborhood?
c. A city?
(15 pts)
Descriptions and color photos are attached at the end of this exercise section for seven types of asphalt pavement distress. Your assignment is to walk around the local area and identify as many of the seven types of pavement surface distress as you can. List a specific location for each type you find, describe the distress, including an estimate of the size of the distressed area, and either take a photograph or produce a sketch of the distress. You may do this problem with one or more colleagues.
Important Note: Watch out for traffic! Work during daylight hours and do not expose yourself to traffic. Parking lots and sidewalks can be used for this exercise!


6.S: An Overview of Organic Reactions (Summary)
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Concepts & Vocabulary
6.1: Kinds of Organic Reactions
- Addition reactions increase the number of sigma bonds in a molecule.
- Elimination reactions reduce the number of sigma bonds in a molecule.
- Substitution reactions incorporate replacement of an atom or group with another.
- Rearrangement reactions cause a molecule to be converted to a constitutional isomer without gaining or losing any atoms.
6.2: How Organic Reactions Occur: Mechanisms
- A reaction mechanism describes movement of electrons by using curved arrows to show bonds that are breaking and forming.
- Homolysis occurs when a bond breaks with each atom keeping one electron.
- Heterolysis occurs when a bond breaks and both electrons remain with one of the atoms.
- Some reactions occur in more than one step with a reactive intermediate formed briefly on the way to the new product.
- Reactive intermediates can be charged species such as carbocations and carbanions or uncharged species such as radicals .
- In organic chemistry Lewis acids are more often referred to as electrophiles , having an affinity for an electron pair.
- In organic chemistry Lewis bases are more often referred to as nucleophiles , having an electron pair that is available to bond to an electrophile .
- Ionic reactions involve charged species.
- Polar reactions involve bonds with unequally shared electrons.
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Radical chain reactions
have three distinct phases: initiation, propagation and termination.
- Initiation causes radicals to be created from non-radical species.
- During the Propagation phase, radicals react with stable molecules to form new radicals.
- Termination occurs when two radicals react together to form a stable molecule.
- Carbon when bonded to a halogen, oxygen, nitrogen, sulfur, or metal has a partial positive charge. This allows these carbons to react with many nucleophiles .
- For carbonyl groups bond polarity is reinforced by resonance making the carbon even more positive than in other molecules. This makes carbonyl groups prone to addition and substitution reactions with nucleophiles .
- Nucleophiles have electron rich atoms that are able to donate a pair of electrons.
- In nucleophilic substitution reactions, the electrophile is typically carbon bonded to a more electronegative atom.
6.5: An Example of a Polar Reaction: Addition of HBr to Ethylene
- Alkene addition reaction with HBr occurs through the pi bond reacting as a nucleophile and abstracting a proton from the acid. This creates a carbocation intermediate which reacts with the bromide ion to form the final product.
- Reaction rates for this alkene addition reaction increase with larger halogens and more substituted alkenes.
- Markovnikov's Rule states that addition reactions of unsymmetrical alkenes yield the more substituted product.
6.6: Using Curved Arrows in Polar Reaction Mechanisms
- Curved arrows in mechanism drawings always represent electrons moving, starting at either a bond or lone pair of electrons.
- Electrons flow from electron rich to electron poor.
6.7: Describing a Reaction: Equilibria, Rates, and Energy Changes
- Exergonic reactions have a negative free energy meaning they are thermodynamically favorable and give off energy.
- Endergonic reactions have a positive free energy and require energy from the surroundings to occur.
6.8: Describing a Reaction: Bond Dissociation Energies
- Bond dissociation energy for a molecule is the difference in enthalpy of formation ( homolytic ) for the products and reactants.
- Bond dissociation energies are independent of path of reaction, so they do not give direct information on mechanisms. However, they can be used to evaluate the results of individual steps of a mechanism.
- Bond dissociation energies show that sigma bonds formed with sp hybridized carbon are stronger than sp 2 which are stronger than bonds formed with sp 3 carbons.
- Bond dissociation energies show that carbon-hydrogen bonds on primary carbons are stronger than secondary, which are stronger than tertiary.
6.9: Describing a Reaction: Energy Diagrams and Transition States
- Reaction coordinate diagrams are a special type of energy diagram that has the reaction coordinate (or reaction progress) on the x-axis.
- Thermodynamics of a reaction is conveyed on a reaction coordinate diagram by the difference in energy between the reactants and products.
- Activation energy is the energy barrier to a reaction occurring.
- A transition state is the highest energy point during the process of bonds forming and breaking in a reaction step.
- Kinetics of a reaction is conveyed on a reaction coordinate diagram by the difference in energy between the reactants and transition state.
- A r ate expression relates rate to the rate constant and concentration of reactants.
6.10: Describing a Reaction: Intermediates
- A reaction intermediate is a short-lived species that goes on to react in a subsequent reaction step.
- Reaction intermediates appear as a local minimum (or valley) on a reaction coordinate diagram.
- Catalysts cause reaction rates to increase by lowing activation energy.
6.11: A Comparison between Biological Reactions and Laboratory Reactions
- An enzyme active site is the location where the enzyme interacts with its substrate and where catalysis occurs.
- Substrates are reactant molecules in enzymatic reactions.
Skills to Master
- Skill 6.1 Identify organic reactions by type (addition, elimination, substitution, rearrangement).
- Skill 6.2 Draw homolytic and heterolytic bond breaking as part of reaction mechanisms.
- Skill 6.3 Identify radical and ionic reactions.
- Skill 6.4 Identify and write out steps in a typical radical substitution reaction (initiation, propagation, termination).
- Skill 6.5 Identify polarity of bonds in organic molecules.
- Skill 6.6 Use curved arrows to indicate movement of electrons in resonance and reaction mechanisms.
- Skill 6.7 Predict whether a chemical species will act as an electrophile or nucleophile.
- Skill 6.8 Write an equilibrium expression for a reaction.
- Skill 6.9 Determine the direction of a reaction based on the equilibrium constant.
- Skill 6.10 Explain how rate and equilibrium are related to \(ΔG^o\) and \(K_{eq}\).
- Skill 6.11 Calculate bond dissociation energy given enthalpies of formation for reactants and products.
- Skill 6.12 Describe order of bond strength based on bond dissociation energy.
- Skill 6.13 Explain activation energy, kinetics, thermodynamics and transition states based on energy diagrams (reaction coordinate diagrams).
- Skill 6.14 Predict possible transition state structures for single reaction steps.
- Skill 6.15 Differentiate between transition states and intermediates.
- Skill 6.16 Draw a reaction coordinate diagram for a given multi-step process.
- Skill 6.17 Interpret a reaction coordinate diagram for a multi-step process.
- Skill 6.18 Briefly explain how enzymes catalyze reactions.
Memorization Tasks
MT 6.1 Memorize that arrows in reaction mechanisms always define movement of electrons.
MT 6.2 Memorize the relative electronegativities of common atoms (necessary for determining polarity of bonds).
MT 6.3 Memorize the equations that relate equilibrium, free energy, enthalpy and entropy.
ΔGº=–RTlnK
ΔGº=ΔHº–TΔSº