EXERCISES Section 5.1 (1-21)

1

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let

• $X$ denote the number of hoses being used on the self-service island at a particular time,
• $Y$ denote the number of hoses on the full-service island in use at that time.

The joint pmf of $X$ and $Y$ appears in the accompanying tabulation.

p(x, y)	y=0	y=1	y=2
x = 0	0.10	0.04	0.02
x = 1	0.08	0.20	0.06
x = 2	0.06	0.14	0.30

a. What is $P\left(X=1\text{ and }Y=1\right)$ ?

b. Compute $P\left(X\le 1\text{ and }Y\le 1\right)$.

c. Give a word description of the event $\{X\ne 0$ and $Y\ne 0\}$, and compute the probability of this event.

d. Compute the marginal pmf of $X$ and of $Y$. Using $p_X\left(x\right)$, what is $P\left(X\le 1\right)$ ?

e. Are $X$ and $Y$ independent rv’s? Explain.

2

A large but sparsely populated county has two small hospitals, one at the south end of the county and the other at the north end. The south hospital’s emergency room has four beds, whereas the north hospital’s emergency room has only three beds. Let

• $X$ denote the number of south beds occupied at a particular time on a given day,
• $Y$ denote the number of north beds occupied at the same time on the same day.

Suppose that these two rv’s are independent;

• the pmf of $X$ puts probability masses $.1,.2,.3,.2$, and .2 on the $x$ values $0,1,2,3$, and 4, respectively;
• the pmf of $Y$ distributes probabilities $.1,.3,.4$, and .2 on the $y$ values $0,1,2$, and 3, respectively.

a. Display the joint pmf of $X$ and $Y$ in a joint probability table.

b. Compute $P\left(X\le 1\text{ and }Y\le 1\right)$ by adding probabilities from the joint pmf, and verify that this equals the product of $P\left(X\le 1\right)$ and $P\left(Y\le 1\right)$.

c. Express the event that the total number of beds occupied at the two hospitals combined is at most 1 in terms of $X$ and $Y$, and then calculate this probability.

d. What is the probability that at least one of the two hospitals has no beds occupied?

3

A certain market has both an express checkout line and a superexpress checkout line. Let $X_1$ denote the number of customers in line at the express checkout at a particular time of day, and let $X_2$ denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of $X_1$ and $X_2$ is as given in the accompanying table.

		0	1	2	3
$x_1$	0	0.08	0.07	0.04	0.00
	1	0.06	0.15	0.05	0.04
	2	0.05	0.04	0.10	0.06
	3	0.00	0.03	0.04	0.07
	4	0.00	0.01	0.05	0.06

a. What is $P\left(X_1=1,X_2=1\right)$, that is, the probability that there is exactly one customer in each line?

b. What is $P\left(X_1=X_2\right)$, that is, the probability that the numbers of customers in the two lines are identical?

c. Let $A$ denote the event that there are at least two more customers in one line than in the other line. Express $A$ in terms of $X_1$ and $X_2$, and calculate the probability of this event.

d. What is the probability that the total number of customers in the two lines is exactly four? At least four?

4

Return to the situation described in Exercise 3.

a. Determine the marginal pmf of $X_1$, and then calculate the expected number of customers in line at the express checkout.

b. Determine the marginal pmf of $X_2$.

c. By inspection of the probabilities $P\left(X_1=4\right)$, $P\left(X_2=0\right)$, and $P\left(X_1=4,X_2=0\right)$, are $X_1$ and $X_2$ independent random variables? Explain.

5

The number of customers waiting for gift-wrap service at a department store is an rv $X$ with possible values $0$, $1$, $2$, $3$, $4$ and corresponding probabilities.1,.2,.3,.25,.15. A randomly selected customer will have 1,2, or 3 packages for wrapping with probabilities $.6,.3$, and.1, respectively. Let $Y=$ the total number of packages to be wrapped for the customers waiting in line (assume that the number of packages submitted by one customer is independent of the number submitted by any other customer).

a. Determine $P\left(X=3,Y=3\right)$, i.e., $p\left(3,3\right)$.

b. Determine $p\left(4,11\right)$.

6

Let $X$ denote the number of Canon SLR cameras sold during a particular week by a certain store. The pmf of $X$ is

$x$	0	1	2	3	4
$p_X(x)$	0.1	0.2	0.3	0.25	0.15

Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let $Y$ denote the number of purchasers during this week who buy an extended warranty.

a. What is $P\left(X=4,Y=2\right)$ ? [Hint: This probability equals $P\left(Y=2\mid X=4\right)\cdot P\left(X=4\right)$ ; now think of the four purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended warranty.]

b. Calculate $P\left(X=Y\right)$.

c. Determine the joint pmf of $X$ and $Y$ and then the marginal pmf of $Y$.

7

The joint probability distribution of the number $X$ of cars and the number $Y$ of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.

$p(x,y)$	0	1	2
0	0.025	0.015	0.010
1	0.050	0.030	0.020
2	0.125	0.075	0.050
3	0.150	0.090	0.060
4	0.100	0.060	0.040
5	0.050	0.030	0.020

a. What is the probability that there is exactly one car and exactly one bus during a cycle?

b. What is the probability that there is at most one car and at most one bus during a cycle?

c. What is the probability that there is exactly one car during a cycle? Exactly one bus?

d. Suppose the left-turn lane is to have a capacity of five cars, and that one bus is equivalent to three cars. What is the probability of an overflow during a cycle?

e. Are $X$ and $Y$ independent rv’s? Explain.

8

A stockroom currently has 30 components of a certain type, of which 8 were provided by supplier 1,10 by supplier 2, and 12 by supplier 3. Six of these are to be randomly selected for a particular assembly. Let $X=$ the number of supplier 1’s components selected, $Y=$ the number of supplier 2’s components selected, and $p\left(x,y\right)$ denote the joint pmf of $X$ and $Y$.

a. What is $p\left(3,2\right)$ ? [Hint: Each sample of size 6 is equally likely to be selected. Therefore, $p\left(3,2\right)=$ (number of outcomes with $X=3$ and $Y=2)/($ total number of outcomes). Now use the product rule for counting to obtain the numerator and denominator.]

b. Using the logic of part (a), obtain $p\left(x,y\right)$. (This can be thought of as a multivariate hypergeometric distribution-sampling without replacement from a finite population consisting of more than two categories.)

9

Each front tire on a particular type of vehicle is supposed to be filled to a pressure of $26\mathrm{psi}$. Suppose the actual air pressure in each tire is a random variable

• $X$ for the right tire
• $Y$ for the left tire,

with joint pdf

$$f\left(x,y\right)=\{\begin{array}{cc}K\left(x^2+y^2\right)& 20\le x\le 30,20\le y\le 30\\ 0& \text{ otherwise }\end{array}$$

a. What is the value of $K$ ?

b. What is the probability that both tires are underfilled?

c. What is the probability that the difference in air pressure between the two tires is at most 2 psi?

d. Determine the (marginal) distribution of air pressure in the right tire alone.

e. Are $X$ and $Y$ independent rv’s?

10

Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M. for dinner at a local health-food restaurant. Let $X=$ Annie’s arrival time and $Y=$ Alvie’s arrival time. Suppose $X$ and $Y$ are independent with each uniformly distributed on the interval $\left[5,6\right]$.

a. What is the joint pdf of $X$ and $Y$ ?

b. What is the probability that they both arrive between 5:15 and 5:45?

c. If the first one to arrive will wait only $10\min$ before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is $A=\{\left(x,y\right):\left|x-y\right|\le 1/6\}$.]

11

Two different professors have just submitted final exams for duplication. Let $X$ denote the number of typographical errors on the first professor’s exam and $Y$ denote the number of such errors on the second exam. Suppose $X$ has a Poisson distribution with parameter ${\mu }_1,Y$ has a Poisson distribution with parameter ${\mu }_2$, and $X$ and $Y$ are independent.

a. What is the joint pmf of $X$ and $Y$ ?

b. What is the probability that at most one error is made on both exams combined?

c. Obtain a general expression for the probability that the total number of errors in the two exams is $m$ (where $m$ is a nonnegative integer). Hint: $A=$ $\{\left(x,y\right):x+y=m\}=\{\left(m,0\right),\left(m-1,1\right),\dots$, $\left(1,m-1\right),\left(0,m\right)\}$. Now sum the joint pmf over $\left(x,y\right)\in A$ and use the binomial theorem, which says that

$$\sum _{k=0}^m\left(\begin{array}{c}m\\ k\end{array}\right)a^k{b}^{m-k}={\left(a+b\right)}^m$$

for any $a,b$.

12

Two components of a minicomputer have the following joint pdf for their useful lifetimes $X$ and $Y$ :

$$f\left(x,y\right)=\{\begin{array}{cc}x{e}^{-x\left(1+y\right)}& x\ge 0\text{ and }y\ge 0\\ 0& \text{ otherwise }\end{array}$$

a. What is the probability that the lifetime $X$ of the first component exceeds 3 ?

b. What are the marginal pdf’s of $X$ and $Y$ ? Are the two lifetimes independent? Explain.

c. What is the probability that the lifetime of at least one component exceeds 3 ?

13

You have two lightbulbs for a particular lamp. Let $X=$ the lifetime of the first bulb and $Y=$ the lifetime of the second bulb (both in 1000s of hours). Suppose that $X$ and $Y$ are independent and that each has an exponential distribution with parameter $\lambda =1$.

a. What is the joint pdf of $X$ and $Y$ ?

b. What is the probability that each bulb lasts at most 1000 hours (i.e., $X\le 1$ and $Y\le 1$ )?

c. What is the probability that the total lifetime of the two bulbs is at most 2 ? [Hint: Draw a picture of the region $A=\{\left(x,y\right):x\ge 0,y\ge 0,x+y\le 2\}$ before integrating.]

d. What is the probability that the total lifetime is between 1 and 2 ?

14

Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the other lifetimes, and that each lifetime has an exponential distribution with parameter $\lambda$.

a. What is the probability that all ten bulbs fail before time $t$ ?

b. What is the probability that exactly $k$ of the ten bulbs fail before time $t$ ?

c. Suppose that nine of the bulbs have lifetimes that are exponentially distributed with parameter $\lambda$ and that the remaining bulb has a lifetime that is exponentially distributed with parameter $\theta$ (it is made by another manufacturer). What is the probability that exactly five of the ten bulbs fail before time $t$ ?

15

Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component 2 or component 3 functions. Let $X_1,X_2$, and $X_3$ denote the lifetimes of components $1,2$, and 3, respectively. Suppose the $X_i$ ’s are independent of one another and each $X_i$ has an exponential distribution with parameter $\lambda$.

a. Let $Y$ denote the system lifetime. Obtain the cumulative distribution function of $Y$ and differentiate to obtain the pdf. [Hint: $F\left(y\right)=P\left(Y\le y\right)$ ; express the event $\{Y\le y\}$ in terms of unions and/or intersections of the three events $\left\{X_1\le y\right\},\left\{X_2\le y\right\}$, and $\left\{X_3\le y\right\}\text{.}]$

b. Compute the expected system lifetime.

16

a. For $f\left(x_1,x_2,x_3\right)$ as given in Example 5.10, compute the joint marginal density function of $X_1$ and $X_3$ alone (by integrating over $x_2$ ).

b. What is the probability that rocks of types 1 and 3 together make up at most $50\%$ of the sample? [Hint: Use the result of part (a).]

c. Compute the marginal pdf of $X_1$ alone. [Hint: Use the result of part (a).]

17

An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let $X=$ the $x$ coordinate of the point selected and $Y=$ the $y$ coordinate of the point selected. If the circle is centered at $\left(0,0\right)$ and has radius $R$, then the joint pdf of $X$ and $Y$ is

$$f\left(x,y\right)=\{\begin{array}{cc}\frac{1}{\pi R^2}& x^2+y^2\le R^2\\ 0& \text{ otherwise }\end{array}$$

a. What is the probability that the selected point is within $R/2$ of the center of the circular region? [Hint: Draw a picture of the region of positive density $D$. Because $f\left(x,y\right)$ is constant on $D$, computing a probability reduces to computing an area.]

b. What is the probability that both $X$ and $Y$ differ from 0 by at most $R/2$ ?

c. Answer part (b) for $R/\sqrt{2}$ replacing $R/2$.

d. What is the marginal pdf of $X$ ? Of $Y$ ? Are $X$ and $Y$ independent?

18

Refer to Exercise 1 and answer the following questions:

a. Given that $X=1$, determine the conditional pmf of $Y$

• i.e., $p_{Y\mid X}\left(0\mid 1\right)$, $p_{Y\mid X}\left(1\mid 1\right)$, and $p_{Y\mid X}\left(2\mid 1\right)$.

b. Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

c. Use the result of part (b) to calculate the conditional probability $P\left(Y\le 1\mid X=2\right)$.

d. Given that two hoses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island?

19

The joint pdf of pressures for right and left front tires is given in Exercise 9.

a. Determine the conditional pdf of $Y$ given that $X=x$ and the conditional pdf of $X$ given that $Y=y$.

b. If the pressure in the right tire is found to be $22\mathrm{psi}$, what is the probability that the left tire has a pressure of at least 25 psi? Compare this to $P\left(Y\ge 25\right)$.

c. If the pressure in the right tire is found to be $22\mathrm{psi}$, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire?

20

Let $X_1,X_2,X_3,X_4,X_5$, and $X_6$ denote the numbers of blue, brown, green, orange, red, and yellow M&M candies, respectively, in a sample of size $n$. Then these $X_i$ ’s have a multinomial distribution. According to the M&M Web site, the color proportions are $p_1=.24,p_2=.13$, $p_3=.16,p_4=.20,p_5=.13$, and $p_6=.14$.

a. If $n=12$, what is the probability that there are exactly two M&Ms of each color?

b. For $n=20$, what is the probability that there are at most five orange candies? [Hint: Think of an orange candy as a success and any other color as a failure.]

c. In a sample of $20\mathrm{M}\&\mathrm{Ms}$, what is the probability that the number of candies that are blue, green, or orange is at least 10 ?

21

Let $X_1,X_2$, and $X_3$ be the lifetimes of components 1,2, and 3 in a three-component system.

a. How would you define the conditional pdf of $X_3$ given that $X_1=x_1$ and $X_2=x_2$ ?

b. How would you define the conditional joint pdf of $X_2$ and $X_3$ given that $X_1=x_1$ ?