6.5: The Method of Moment-Generating Functions

(The proof of Theorem 6.1 is beyond the scope of this text.)

Exercises

6.37

Let $Y_1, Y_2, ..., Y_n$ independent and identically distributed random variables such that for $0 < p < 1$, $P(Y_i = 1) = p$ and $P(Y_i = 0) = q = 1 - p$. (Such random variables are called Bernoulli random variables.)

a) Find the moment-generating function for the Bernoulli random variable $Y_1$.

b) Find the moment-generating function for $W = Y_1 + Y_2 + ... + Y_n$.

c) What is the distribution of $W$?

6.38

Let $Y_1$ and $Y_2$ be independent random variables with moment-generating functions $m_{Y_1}(t)$ and $m_{Y_2}(t)$, respectively. If $a_1$ and $a_2$ are constants, and $U = a_1 Y_1 + a_2 Y_2$ show that the moment-generating function for $U$ is $m_U(t) = m_{Y_1}(a_1 t) \times m_{Y_2}(a_2 t)$.

6.39

In Exercises 6.11 and 6.25, we considered two electronic components that operate independently, each with a life length governed by the exponential distribution with mean $1$. Use the method of moment-generating functions to obtain the density function for the average life length of the two components.

6.40

Suppose that $Y_1$ and $Y_2$ are independent, standard normal random variables. Find the density function of $U = Y_1^2 + Y_2^2$.

6.41

Let $Y_1, Y_2, ..., Y_n$ be independent, normal random variables, each with mean $\mu$ and variance $\sigma^2$. Let $a_1, a_2, ..., a_n$ denote known constants. Find the density function of the linear combination $U = \sum_{i = 1}^n a_i Y_i$.

6.42

A type of elevator has a maximum weight capacity $Y_1$, which is normally distributed with mean $5000$ pounds and standard deviation $300$ pounds. For a certain building equipped with this type of elevator, the elevator's load, $Y_2$, is a normally distributed random variable with mean $4000$ pounds and standard deviation $400$ pounds. For any given time that the elevator is in use, find the probability that it will be overloaded, assuming that $Y_1$ and $Y_2$ are independent.

6.43

Refer to Exercise 6.41. Let $Y_1, Y_2, ..., Y_n$ be independent, normal random variables, each with mean $\mu$ and variance $\sigma^2$.

a) Find the density function of $\overline{Y} = \frac{1}{n} \sum_{i = 1}^n Y_i$.

b) If $\sigma^2 = 16$ and $n = 25$, what is the probability that the sample mean, $\overline{Y}$, takes on a value that is within one unit of the population mean, $\mu$? That is, find $P(|\overline{Y} - \mu| \leq 1)$.

c) If $\sigma^2 = 16$, find $P(|\overline{Y} - \mu| \leq 1)$ if $n = 36$, $n = 64$, and $n = 81$. Interpret the results of your calculations.

6.44

The weight (in pounds) of "medium-size" watermelons is normally distributed with mean $15$ and variance $4$. A packing container for several melons has a nominal capacity of $140$ pounds. What is the maximum number of melons that should be placed in a single packing container if the nominal weight limit is to be exceeded only $5\%$ of the time? Give reasons for your answer.

6.45

The manager of a construction job needs to figure prices carefully before submitting a bid. He also needs to account for uncertainty (variability) in the amounts of products he might need. To oversimplify the real situation, suppose that a project manager treats the amount of sand, in yards, needed for a construction project as a random variable $Y_1$, which is normally distributed with mean $10$ yards and standard deviation $.5$ yard. The amount of cement mix needed, in hundreds of pounds, is a random variable $Y_2$, which is normally distributed with mean $4$ and standard deviation $.2$. The sand costs $$7$ per yard, and the cement mix costs $$3$ per hundred pounds. Adding $$100$ for other costs, he computes his total cost to be

$$U = 100 + 7Y_1 + 3Y_2.$$

If $Y_1$ and $Y_2$ are independent, how much should the manager bid to ensure that the true costs will exceed the amount bid with a probability of only $.01$? Is the independence assumption reasonable here?

6.46

Suppose that $Y$ has a gamma distribution with $\alpha = n / 2$ for some positive integer $n$ and $\beta$ equal to some specified value. Use the method of moment-generating functions to show that $W = 2Y / \beta$ has a $\chi^2$ distribution with $n$ degrees of freedom.

6.47

A random variable $Y$ has a gamma distribution with $\alpha = 3.5$ and $\beta = 4.2$. Use the result in Exercise 6.46 and the percentage points for the $\chi^2$ distributions given in Table 6, Appendix 3, to find $P(Y > 33.627)$.

6.48

In a missile-testing program, one random variable of interest is the distance between the point at which the missile lands and the center of the target at which the missile was aimed. If we think of the center of the target as the origin of a coordinate system, we can let $Y_1$ denote the north-south distance between the landing point and the target center and let $Y_2$ denote the corresponding east-west distance. (Assume that north and east define positive directions.) The distance between the landing point and the target center is then $U = \sqrt{Y_1^2 + Y_2^2}$. If $Y_1$ and $Y_2$ are independent, standard normal random variables, find the probability density function for $U$.

6.49

Let $Y_1$ be a binomial random variable with $n_1$ trials and probability of success given by $p$. Let $Y_2$ be another binomial random variable with $n_2$ trials and probability of success also given by $p$. If $Y_1$ and $Y_2$ are independent, find the probability function of $Y_1 + Y_2$.

6.50

Let $Y$ be a binomial random variable with $n$ trials and probability of success given by $p$. Show that $n - Y$ is a binomial random variable with $n$ trials and probability of success given by $1 - p$.

6.51

Let $Y_1$ be a binomial random variable with $n_1$ trials and $p_1 = .2$ and $Y_2$ be an independent binomial random variable with $n_2$ trials and $p_2 = .8$. Find the probability function of $Y_1 + n_2 - Y_2$.

6.52

Let $Y_1$ and $Y_2$ be independent Poisson random variables with means $\lambda_1$ and $\lambda_2$, respectively. Find the

a) probability function of $Y_1 + Y_2$.

b) conditional probability function of $Y_1$, given that $Y_1 + Y_2 = m$.

6.53

Let $Y_1, Y_2, ..., Y_n$ be independent binomial random variables with $n_i$ trials probability of success given by $p_i$, $i = 1, 2, ..., n$.

a) If all of the $n_i$'s are equal and all of the $p_i$'s are equal, find the distribution of $\sum_{i = 1}^n Y_i$.

b) If all of the $n_i$'s are different and all of the $p_i$'s are equal, find the distribution of $\sum_{i = 1}^n Y_i$.

c) If all of the $n_i$'s are different and all of the $p_i$'s are equal, find the conditional distribution of $Y_1$ given $\sum_{i = 1}^n Y_i = m$.

d) If all of the $n_i$'s are different and all of the $p_i$'s are equal, find the conditional distribution of $Y_1 + Y_2$ given $\sum_{i = 1}^n Y_i = m$.

e) If all of the $p_i$'s are different, does the method of moment-generating functions work well to find the distribution of $\sum_{i = 1}^n Y_i$? Why?

6.54

Let $Y_1, Y_2, ..., Y_n$ be independent Poisson random variables with means $\lambda_1, \lambda_2, ..., \lambda_n$, respectively. Find the

a) probability function of $\sum_{i = 1}^n Y_i$.

b) conditional probability function of $Y_1$, given that $\sum_{i = 1}^n Y_i = m$.

c) conditional probability function of $Y_1 + Y_2$, given that $\sum_{i = 1}^n Y_i = m$.

6.55

Customers arrive at a department store checkout counter according to a Poisson distribution with a mean of $7$ per hour. In a given two-hour period, what is the probability that $20$ or more customers will arrive at the counter?

6.56

The length of time necessary to tune up a car is exponentially distributed with a mean of $.5$ hour. If two cars are waiting for a tune-up and the service times are independent, what is the probability that the total time for the two tune-ups will exceed $1.5$ hours? [Hint: Recall the result of Example 6.12.]

6.57

Let $Y_1, Y_2, ..., Y_n$ be independent random variables such that each $Y_i$ has a gamma distribution with parameters $\alpha_i$ and $\beta$. That is, the distributions of the $Y_i$'s might have different $\alpha$'s, but all have the same value for $\beta$. Prove that $U = Y_1 + Y_2 + ... + Y_n$ has a gamma distribution with parameters $\alpha_1 + \alpha_2 + ... + \alpha_n$ and $\beta$.

6.58

We saw in Exercise 5.159 that the negative binomial random variable $Y$ can be written as $Y = \sum_{i = 1}^r W_i$, where $W_1, W_2, ..., W_r$ are independent geometric random variables with parameter $p$.

a) Use this fact to derive the moment-generating function for $Y$.

b) Use the moment-generating function to show that $E(Y) = r / p$ and $V(Y) = r(1 - p) / p^2$.

c) Find the conditional probability function for $W_1$, given that $Y = W_1 + W_2 + ... + W_r = m$.

6.59

Show that if $Y_1$ has a $\chi^2$ distribution with $\nu_1$ degrees of freedom and $Y_2$ has a $\chi^2$ distribution with $\nu_2$ degrees of freedom, then $U = Y_1 + Y_2$ has a $\chi^2$ distribution with $\nu_1 + \nu_2$ degrees of freedom, provided that $Y_1$ and $Y_2$ are independent.

6.60

Suppose that $W = Y_1 + Y_2$ where $Y_1$ and $Y_2$ are independent. If $W$ has a $\chi^2$ distribution with $\nu$ degrees of freedom and $Y_1$ has a $\chi^2$ distribution with $\nu_1 < \nu$ degrees of freedom, show that $Y_2$ has a $\chi^2$ distribution with $\nu - \nu_1$ degrees of freedom.

6.61

Refer to Exercise 6.52. Suppose that $W = Y_1 + Y_2$ where $Y_1$ and $Y_2$ are independent. If $W$ has a Poisson distribution with mean $\lambda$ and $Y_1$ has a Poisson distribution with mean $\lambda_1 < \lambda$, show that $Y_2$ has a Poisson distribution with mean $\lambda - \lambda_1$.

6.62

Let $Y_1$ and $Y_2$ be independent normal random variables, each with mean $0$ and variance $\sigma^2$. Define $U_1 = Y_1 + Y_2$ and $U_2 = Y_1 - Y_2$. Show that $U_1$ and $U_2$ are independent normal random variables, each with mean $0$ and variance $2 \sigma^2$. [Hint: If $(U_1 , U_2)$ has a joint moment-generating function $m(t_1, t_2)$, then $U_1$ and $U_2$ are independent if and only if $m(t_1, t_2) = m_{U_1}(t_1) m_{U_2}(t_2)$.]