EXERCISES Section 3.3 (29-45)

29

The pmf of the amount of memory $X\left(\mathrm{GB}\right)$ in a purchased flash drive was given in Example 3.13 as

x	1	2	4	8	16
p(x)	.05	.10	.35	.40	.10

Compute the following:

a. $E\left(X\right)$

b. $V\left(X\right)$ directly from the definition

c. The standard deviation of $X$

d. $V\left(X\right)$ using the shortcut formula

30

An individual who has automobile insurance from a certain company is randomly selected. Let $Y$ be the number of moving violations for which the individual was cited during the last 3 years. The pmf of $Y$ is

y	0	1	2	3
p(y)	.60	.25	.10	.05

a. Compute $E\left(Y\right)$ .

b. Suppose an individual with $Y$ violations incurs a surcharge of \ {100}{Y}^{2}$ . Calculate the expected amount of the surcharge.

31

Refer to Exercise 12 and calculate $V\left(Y\right)$ and ${\sigma }_Y$ . Then determine the probability that $Y$ is within 1 standard deviation of its mean value.

32

A certain brand of upright freezer is available in three different rated capacities: $16{\mathrm{ft}}^3$, $18{\mathrm{ft}}^3$, and $20{\mathrm{ft}}^3$ . Let $X=$ the rated capacity of a freezer of this brand sold at a certain store. Suppose that $X$ has pmf

$x$	16	18	20
$p(x)$	.2	.5	.3

a. Compute $E\left(X\right),E\left(X^2\right)$ , and $V\left(X\right)$ .

b. If the price of a freezer having capacity $X$ is $70X-650$ , what is the expected price paid by the next customer to buy a freezer?

c. What is the variance of the price paid by the next customer?

d. Suppose that although the rated capacity of a freezer is $X$ , the actual capacity is $h\left(X\right)=X-.008X^2$ . What is the expected actual capacity of the freezer purchased by the next customer?

33

Let $X$ be a Bernoulli rv with pmf as in Example 3.18.

a. Compute $E\left(X^2\right)$ .

b. Show that $V\left(X\right)=p\left(1-p\right)$ .

c. Compute $E\left(X^{79}\right)$ .

34

Suppose that the number of plants of a particular type found in a rectangular sampling region (called a quadrat by ecologists) in a certain geographic area is an rv $X$ with pmf $p\left(x\right)=\{\begin{array}{ll}\mathrm{c}/x^3& x=1,2,3,\dots \\ 0& \text{ otherwise }\end{array}$

Is $E\left(X\right)$ finite? Justify your answer (this is another distribution that statisticians would call heavy-tailed).

35

A small market orders copies of a certain magazine for its magazine rack each week. Let $X=$ demand for the magazine, with pmf

$x$	1	2	3	4	5	6
$p(x)$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{3}{15}$	$\frac{4}{15}$	$\frac{3}{15}$	$\frac{2}{15}$

Suppose the store owner actually pays \ {2.00}$foreachcopyofthemagazineandthepricetocustomersis$$ {4.00}$ . If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine?

Hint: For both three and four copies ordered, express net revenue as a function of demand $X$ , and then compute the expected revenue.

36

Let $X$ be the damage incurred (in \$$ ) in a certain type of accident during a given year. Possible X$valuesare0,${1000}$,${5000}, and 10000, with probabilities .8,.1,.08, and .02 , respectively. A particular company offers a \500 deductible policy. If the company wishes its expected profit to be \ {100}$ , what premium amount should it charge?

37

The $n$ candidates for a job have been ranked $1,2,3,\dots ,n$ . Let $X=$ the rank of a randomly selected candidate, so that $X$ has pmf $p\left(x\right)=\{\begin{array}{ll}1/n& x=1,2,3,\dots ,n\\ 0& \text{ otherwise }\end{array}$ (this is called the discrete uniform distribution). Compute $E\left(X\right)$ and $V\left(X\right)$ using the shortcut formula.

Hint: The sum of the first $n$ positive integers is $n\left(n+1\right)/2$ , whereas the sum of their squares is $n\left(n+1\right)\left(2n+1\right)/6$ .

38

Possible values of $X$ , the number of components in a system submitted for repair that must be replaced, are 1 , 2,3, and 4 with corresponding probabilities $.15,.35,.35$ , and .15, respectively.

a. Calculate $E\left(X\right)$ and then $E\left(5-X\right)$ .

b. Would the repair facility be better off charging a flat fee of \ {75}$orelsetheamount$$ \left\lbrack {{150}/\left( {5 - X}\right) }\right\rbrack$?Note:Itisnotgenerallytruethat$E\left( {c/Y}\right) = c/E\left( Y\right)$ .

39

A chemical supply company currently has in stock $100\mathrm{lb}$ of a certain chemical, which it sells to customers in 5-lb batches. Let $X=$ the number of batches ordered by a randomly chosen customer, and suppose that $X$ has pmf

$x$	1	2	3	4
$p(x)$	.2	.4	.3	.1

Compute $E\left(X\right)$ and $V\left(X\right)$ . Then compute the expected number of pounds left after the next customer’s order is shipped and the variance of the number of pounds left.

Hint: The number of pounds left is a linear function of $X$ .

40

a. Draw a line graph of the pmf of $X$ in Exercise 35. Then determine the pmf of $-X$ and draw its line graph. From these two pictures, what can you say about $V\left(X\right)$ and $V\left(-X\right)$ ?

b. Use the proposition involving $V\left(aX+b\right)$ to establish a general relationship between $V\left(X\right)$ and $V\left(-X\right)$ .

41

Use the definition in Expression (3.13) to prove that $V\left(aX+b\right)=a^2\cdot {\sigma }_X^2$ . Hint: With $h\left(X\right)=aX+b$ , $E\left[h\left(X\right)\right]=a\mu +b$ where $\mu =E\left(X\right)$ .

42

Suppose $E\left(X\right)=5$ and $E\left[X\left(X-1\right)\right]=27.5$ . What is

a. $E\left(X^2\right)$ ? [Hint: First verify that $E\left[X\left(X-1\right)\right]=E\left(X^2\right)-E\left(X\right)]$ ?

b. $V\left(X\right)$ ?

c. The general relationship among the quantities $E\left(X\right)$ , $E\left[X\left(X-1\right)\right]$ , and $V\left(X\right)$ ?

43

Write a general rule for $E\left(X-c\right)$ where $c$ is a constant. What happens when $c=\mu$ , the expected value of $X$ ?

44

A result called Chebyshev’s inequality states that for any probability distribution of an $\mathrm{rv}X$ and any number $k$ that is at least $1$, $P\left(\left|X-\mu \right|\ge k\sigma \right)\le 1/k^2$. In words, the probability that the value of $X$ lies at least $k$ standard deviations from its mean is at most $1/k^2$ .

a. What is the value of the upper bound for $k=2$? $k=3$? $k=4$? $k=5$? $k=10$?

b. Compute $\mu$ and $\sigma$ for the distribution of Exercise 13. Then evaluate $P\left(\left|X-\mu \right|\ge k\sigma \right)$ for the values of $k$ given in part (a). What does this suggest about the upper bound relative to the corresponding probability?

c. Let $X$ have possible values $-1,0$ , and 1, with probabilities $\frac{1}{18},\frac{8}{9}$ , and $\frac{1}{18}$ , respectively. What is $P\left(\left|X-\mu \right|\ge 3\sigma \right)$ , and how does it compare to the corresponding bound?

d. Give a distribution for which $P\left(\left|X-\mu \right|\ge 5\sigma \right)=.04$ .

45

If $a\le X\le b$ , show that $a\le E\left(X\right)\le b$ .