Introduction to Mathematical Statistics

Robert V. Hogg, Joseph W. McKean, Allen T. Craig

Chapter 3

Some Special Distributions - all with Video Answers

Educators

Section 1

The Binomial and Related Distributions

Problem 1

If the mgf of a random variable $X$ is $\left(\frac{1}{3}+\frac{2}{3} e^{t}\right)^{5}$, find $P(X=2$ or 3$)$. Verify using the $\mathrm{R}$ function dbinom.

Manik Pulyani

Manik Pulyani

Numerade Educator

Problem 2

The mgf of a random variable $X$ is $\left(\frac{2}{3}+\frac{1}{3} e^{t}\right)^{9}$.
(a) Show that
$$
P(\mu-2 \sigma<X<\mu+2 \sigma)=\sum_{x=1}^{5}\left(\begin{array}{l}
9 \\
x
\end{array}\right)\left(\frac{1}{3}\right)^{x}\left(\frac{2}{3}\right)^{9-x}
$$
(b) Use $\mathrm{R}$ to compute the probability in Part (a).

Manik Pulyani

Numerade Educator

Problem 3

If $X$ is $b(n, p)$, show that
$$
E\left(\frac{X}{n}\right)=p \quad \text { and } \quad E\left[\left(\frac{X}{n}-p\right)^{2}\right]=\frac{p(1-p)}{n}
$$

Ameer Said

Numerade Educator

Problem 4

Let the independent random variables $X_{1}, X_{2}, \ldots, X_{40}$ be iid with the common pdf $f(x)=3 x^{2}, 0<x<1$, zero elsewhere. Find the probability that at least 35 of the $X_{i}$ 's exceed $\frac{1}{2}$.

Manik Pulyani

Numerade Educator

Problem 5

Over the years, the percentage of candidates passing an entrance exam to a prestigious law school is $20 \%$. At one of the testing centers, a group of 50 candidates take the exam and 20 pass. Is this odd? Answer on the basis that $X \geq 20$ where $X$ is the number that pass in a group of 50 when the probability of a pass is $0.2$.

Manik Pulyani

Numerade Educator

Problem 6

Let $Y$ be the number of successes throughout $n$ independent repetitions of a random experiment with probability of success $p=\frac{1}{4}$. Determine the smallest value of $n$ so that $P(1 \leq Y) \geq 0.70$

Manik Pulyani

Numerade Educator

Problem 7

Let the independent random variables $X_{1}$ and $X_{2}$ have binomial distribution with parameters $n_{1}=3, p=\frac{2}{3}$ and $n_{2}=4, p=\frac{1}{2}$, respectively. Compute $P\left(X_{1}=X_{2}\right)$
Hint: List the four mutually exclusive ways that $X_{1}=X_{2}$ and compute the probability of each.

Manik Pulyani

Numerade Educator

Problem 8

For this exercise, the reader must have access to a statistical package that obtains the binomial distribution. Hints are given for $\mathrm{R}$ code, but other packages can be used too.
(a) Obtain the plot of the pmf for the $b(15,0.2)$ distribution. Using $\mathrm{R}$, the following commands return the plot:
$x<-0: 15 ;$ plot $\left(\operatorname{dbinom}(x, 15, .2)^{-} x\right)$
(b) Repeat part (a) for the binomial distributions with $n=15$ and with $p=$ $0.10,0.20, \ldots, 0.90 .$ Comment on the shapes of the pmf's as $p$ increases. Use the following $\mathrm{R}$ segment:
$\mathrm{x}<-0: 15 ; \quad$ par $(\mathrm{mfrow}=\mathrm{c}(3,3)) ; \mathrm{p}<-1: 9 / 10$
for $(j$ in $p)\left\{\right.$ plot $\left(\right.$ dbinom $\left.(x, 15, j)^{\sim} x\right) ;$ title(paste $\left.\left.(" p=", j)\right)\right\}$
(c) Let $Y=\frac{X}{n}$, where $X$ has a $b(n, 0.05)$ distribution. Obtain the plots of the pmfs of $Y$ for $n=10,20,50,200 .$ Comment on the plots (what do the plots seem to be converging to as $n$ gets large? ).

Manik Pulyani

Numerade Educator

Problem 9

If $x=r$ is the unique mode of a distribution that is $b(n, p)$, show that
$$
(n+1) p-1<r<(n+1) p
$$
This substantiates the comments made in Part (b) of Exercise 3.1.8. Hint: Determine the values of $x$ for which $p(x+1) / p(x)>1$.

Manik Pulyani

Numerade Educator

Problem 10

Suppose $X$ is $b(n, p)$. Then by definition the pmf is symmetric if and only if $p(x)=p(n-x)$, for $x=0, \ldots, n$. Show that the pmf is symmetric if and only if $p=1 / 2$

Manik Pulyani

Numerade Educator

Problem 11

Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.

Ameer Said

Numerade Educator

Problem 12

Let $X_{1}, X_{2}, \ldots, X_{k-1}$ have a multinomial distribution.
(a) Find the mgf of $X_{2}, X_{3}, \ldots, X_{k-1}$.
(b) What is the pmf of $X_{2}, X_{3}, \ldots, X_{k-1} ?$
(c) Determine the conditional pmf of $X_{1}$ given that $X_{2}=x_{2}, \ldots, X_{k-1}=x_{k-1}$.
(d) What is the conditional expectation $E\left(X_{1} \mid x_{2}, \ldots, x_{k-1}\right) ?$

Manik Pulyani

Numerade Educator

Problem 13

Let $X$ be $b(2, p)$ and let $Y$ be $b(4, p)$. If $P(X \geq 1)=\frac{5}{9}$, find $P(Y \geq 1)$.

Evelyn Cunningham

Evelyn Cunningham

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Problem 14

Let $X$ have a binomial distribution with parameters $n$ and $p=\frac{1}{3}$. Determine the smallest integer $n$ can be such that $P(X \geq 1) \geq 0.85$.

Ameer Said

Numerade Educator

Problem 15

Let $X$ have the pmf $p(x)=\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^{x}, x=0,1,2,3, \ldots$, zero elsewhere. Find the conditional pmf of $X$ given that $X \geq 3$.

Manik Pulyani

Numerade Educator

Problem 16

One of the numbers $1,2, \ldots, 6$ is to be chosen by casting an unbiased die. Let this random experiment be repeated five independent times. Let the random variable $X_{1}$ be the number of terminations in the set $\{x: x=1,2,3\}$ and let the random variable $X_{2}$ be the number of terminations in the set $\{x: x=4,5\}$. Compute $P\left(X_{1}=2, X_{2}=1\right)$

Manik Pulyani

Numerade Educator

Problem 17

Show that the moment generating function of the negative binomial distribution is $M(t)=p^{r}\left[1-(1-p) e^{t}\right]^{-r}$. Find the mean and the variance of this distribution. Hint: In the summation representing $M(t)$, make use of the negative binomial series. ${ }^{1}$

Manik Pulyani

Numerade Educator

Problem 18

One way of estimating the number of fish in a lake is the following capturerecapture sampling scheme. Suppose there are $N$ fish in the lake where $N$ is unknown. A specified number of fish $T$ are captured, tagged, and released back to the lake. Then at a specified time and for a specified positive integer $r$, fish are captured until the $r t h$ tagged fish is caught. The random variable of interest is $Y$ the number of nontagged fish caught.
(a) What is the distribution of $Y ?$ Identify all parameters.
(b) What is $E(Y)$ and the $\operatorname{Var}(Y)$ ?
(c) The method of moment estimate of $N$ is to set $Y$ equal to the expression for $E(Y)$ and solve this equation for $N .$ Call the solution $\hat{N}$. Determine $\hat{N}$.
(d) Determine the mean and variance of $\hat{N}$.

Manik Pulyani

Numerade Educator

Problem 19

Consider a multinomial trial with outcomes $1,2, \ldots, k$ and respective probabilities $p_{1}, p_{2}, \ldots, p_{k} .$ Let ps denote the $\mathrm{R}$ vector for $\left(p_{1}, p_{2}, \ldots, p_{k}\right) .$ Then a single random trial of this multinomial is computed with the command multitrial (ps), where the required $\mathrm{R}$ functions are: ${ }^{2}$
(a) Compute 10 random trials if $\mathrm{ps}=\mathrm{c}(.3, .2, .2, .2, .1)$.
(b) Compute 10,000 random trials for ps as in (a). Check to see how close the estimates of $p_{i}$ are with $p_{i}$.

Manik Pulyani

Numerade Educator

Problem 20

Using the experiment in part (a) of Exercise 3.1.19, consider a game when a person pays $\$ 5$ to play. If the trial results in a 1 or 2, she receives nothing: if a 3 , she receives $\$ 1$; if a 4 , she receives $\$ 2$; and if a 5 , she receives $\$ 20$. Let $G$ be her gain.
(a) Determine $E(G)$.
(b) Write $\mathrm{R}$ code that simulates the gain. Then simulate it 10,000 times, collecting the gains. Compute the average of these 10,000 gains and compare it with $E(G)$

Manik Pulyani

Numerade Educator

Problem 21

Let $X_{1}$ and $X_{2}$ have a trinomial distribution. Differentiate the momentgenerating function to show that their covariance is $-n p_{1} p_{2}$.

Manik Pulyani

Numerade Educator

Problem 22

If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.

Manik Pulyani

Numerade Educator

Problem 23

Let an unbiased die be cast at random seven independent times. Compute the conditional probability that each side appears at least once given that side 1 appears exactly twice.

Manik Pulyani

Numerade Educator

Problem 24

Compute the measures of skewness and kurtosis of the binomial distribution $b(n, p)$

Manik Pulyani

Numerade Educator

Problem 25

Let
$$
p\left(x_{1}, x_{2}\right)=\left(\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right)\left(\frac{1}{2}\right)^{x_{1}}\left(\frac{x_{1}}{15}\right), \begin{aligned}
&x_{2}=0,1, \ldots, x_{1} \\
&x_{1}=1,2,3,4,5
\end{aligned}
$$
zero elsewhere, be the joint pmf of $X_{1}$ and $X_{2}$. Determine
(a) $E\left(X_{2}\right)$.
(b) $u\left(x_{1}\right)=E\left(X_{2} \mid x_{1}\right)$.
(c) $E\left[u\left(X_{1}\right)\right]$.
Compare the answers of parts (a) and (c). Hint: Note that $E\left(X_{2}\right)=\sum_{x_{1}=1}^{5} \sum_{x_{2}=0}^{x_{1}} x_{2} p\left(x_{1}, x_{2}\right)$

Manik Pulyani

Numerade Educator

Problem 26

Three fair dice are cast. In 10 independent casts, let $X$ be the number of times all three faces are alike and let $Y$ be the number of times only two faces are alike. Find the joint $\mathrm{pmf}$ of $X$ and $Y$ and compute $E(6 X Y)$.

Manik Pulyani

Numerade Educator

Problem 27

Let $X$ have a geometric distribution. Show that
$$
P(X \geq k+j \mid X \geq k)=P(X \geq j)
$$
where $k$ and $j$ are nonnegative integers. Note that we sometimes say in this situation that $X$ is memoryless.

Manik Pulyani

Numerade Educator

Problem 28

Let $X$ equal the number of independent tosses of a fair coin that are required to observe heads on consecutive tosses. Let $u_{n}$ equal the $n$ th Fibonacci number, where $u_{1}=u_{2}=1$ and $u_{n}=u_{n-1}+u_{n-2}, n=3,4,5, \ldots$
(a) Show that the pmf of $X$ is
$$
p(x)=\frac{u_{x-1}}{2^{x}}, \quad x=2,3,4, \ldots
$$
(b) Use the fact that
$$
u_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right]
$$
to show that $\sum_{x=2}^{\infty} p(x)=1$

Manik Pulyani

Numerade Educator

Problem 29

Let the independent random variables $X_{1}$ and $X_{2}$ have binomial distributions with parameters $n_{1}, p_{1}=\frac{1}{2}$ and $n_{2}, p_{2}=\frac{1}{2}$, respectively. Show that $Y=X_{1}-X_{2}+n_{2}$ has a binomial distribution with parameters $n=n_{1}+n_{2}, p=\frac{1}{2}$.

Manik Pulyani

Numerade Educator

Problem 30

Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about $5 \%$ defective items. Let $X$ be the number of defective items in a sample without replacement of size $n=10 .$ Suppose the factory returns the shipment if $X \geq 2$.
(a) Obtain the probability that the factory returns a shipment of items that has $5 \%$ defective items.
(b) Suppose the shipment has $10 \%$ defective items. Obtain the probability that the factory returns such a shipment.
(c) Obtain approximations to the probabilities in parts (a) and (b) using appropriate binomial distributions.
Note: If you do not have access to a computer package with a hypergeometric command, obtain the answer to (c) only. This is what would have been done in practice 20 years ago. If you have access to $\mathrm{R}$, then the command dhyper $(\mathrm{x}, \mathrm{D}, \mathrm{N}-\mathrm{D}, \mathrm{n})$ returns the probability in expression (3.1.7).

Manik Pulyani

Numerade Educator

Problem 31

Show that the variance of a hypergeometric $(N, D, n)$ distribution is given by expression (3.1.8). Hint: First obtain $E[X(X-1)]$ by proceeding in the same way as the derivation of the mean given in Section 3.1.3.

Manik Pulyani

Numerade Educator