EXERCISES Section 5.2 (22-36)

22

An instructor has given a short quiz consisting of two parts. For a randomly selected student, let

• $X=$ the number of points earned on the first part
• $Y=$ the number of points earned on the second part.

Suppose that the joint pmf of $X$ and $Y$ is given in the accompanying table.

$p(x,y)$	0	5	10	15
0	.02	.06	.02	.10
5	.04	.15	.20	.10
10	.01	.15	.14	.01

a. If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score $E\left(X+Y\right)$ ?

b. If the maximum of the two scores is recorded, what is the expected recorded score?

23

The difference between the number of customers in line at the express checkout and the number in line at the super-express checkout in Exercise 3 is $X_1-X_2$. Calculate the expected difference.

24

Six individuals, including A and B, take seats around a circular table in a completely random fashion. Suppose the seats are numbered $1,\dots ,6$. Let $X=\mathrm{A}$ ’s seat number and $Y=$ B’s seat number. If A sends a written message around the table to $\mathrm{B}$ in the direction in which they are closest, how many individuals (including A and B) would you expect to handle the message?

25

A surveyor wishes to lay out a square region with each side having length $L$. However, because of a measurement error, he instead lays out a rectangle in which the north-south sides both have length $X$ and the east-west sides both have length $Y$. Suppose that $X$ and $Y$ are independent and that each is uniformly distributed on the interval $\left[L-A,L+A\right]$ (where $0<A<L)$. What is the expected area of the resulting rectangle?

26

Consider a small ferry that can accommodate cars and buses. The toll for cars is \ 3$,andthetollforbusesis$$ {10}$.Let$X$and$Y$denotethenumberofcarsandbuses,respectively,carriedonasingletrip.Supposethejointdistributionof$X$and$Y$ is as given in the table of Exercise 7. Compute the expected revenue from a single trip.

27

Annie and Alvie have agreed to meet for lunch between noon (0:00 P.M.) and 1:00 P.M. Denote Annie’s arrival time by $X$, Alvie’s by $Y$, and suppose $X$ and $Y$ are independent with pdf’s

$$f_X\left(x\right)=\{\begin{array}{ll}3x^2& 0\le x\le 1\\ 0& \text{otherwise}\end{array}$$ $$f_Y\left(y\right)=\{\begin{array}{ll}2y& 0\le y\le 1\\ 0& \text{otherwise}\end{array}$$

What is the expected amount of time that the one who arrives first must wait for the other person? Hint: $h\left(X,Y\right)=\left|X-Y\right|$.

28

Show that if $X$ and $Y$ are independent rv’s, then $E\left(XY\right)=E\left(X\right)\cdot E\left(Y\right)$. Then apply this in Exercise 25. [Hint: Consider the continuous case with $f\left(x,y\right)=$ $f_X\left(x\right)\cdot f_Y\left(y\right)\text{.}]$

29

Compute the correlation coefficient $\rho$ for $X$ and $Y$ of Example 5.16 (the covariance has already been computed).

30

a. Compute the covariance for $X$ and $Y$ in 22.

b. Compute $\rho$ for $X$ and $Y$ in the same exercise.

31

a. Compute the covariance between $X$ and $Y$ in Exercise 9.

b. Compute the correlation coefficient $\rho$ for this $X$ and $Y$.

32

Reconsider the minicomputer component lifetimes $X$ and $Y$ as described in Exercise 12. Determine $E\left(XY\right)$. What can be said about $\mathrm{Cov}\left(X,Y\right)$ and $\rho$ ?

33

Use the result of Exercise 28 to show that when $X$ and $Y$ are independent, $\mathrm{Cov}\left(X,Y\right)=\mathrm{Corr}\left(X,Y\right)=0$.

34

a. Recalling the definition of ${\sigma }^2$ for a single rv $X$, write a formula that would be appropriate for computing the variance of a function $h\left(X,Y\right)$ of two random variables. [Hint: Remember that variance is just a special expected value.]

b. Use this formula to compute the variance of the recorded score $h\left(X,Y\right)\left[=\max \left(X,Y\right)\right]$ in part (b) of Exercise 22.

35

a. Use the rules of expected value to show that $\mathrm{Cov}\left(aX+b,cY+d\right)=ac\mathrm{Cov}\left(X,Y\right)$.

b. Use part (a) along with the rules of variance and standard deviation to show that $\mathrm{Corr}(aX+b$, $cY+d)=\mathrm{Corr}\left(X,Y\right)$ when $a$ and $c$ have the same sign.

c. What happens if $a$ and $c$ have opposite signs?

36

Show that if $Y=aX+b\left(a\ne 0\right)$, then $\mathrm{Corr}\left(X,Y\right)=+1$ or -1. Under what conditions will $\rho =+1$ ?