26 In the computer sales scenario of 23, E(X) 5 2 and
Example 3.26 In the computer sales scenario of Example 3.23, E(X) 5 2 and
E(X 2 ) 5 (0) 2 (.1) 1 (1) 2 (.2) 1 (2) 2 (.3) 1 (3) 2 (.4) 5 5 so V(X) 5 5 2 (2) 2 51 . The profit function h(X) 5 800X 2 900 then has variance
2 (800) V(X) 5 (640,000)(1) 5 640,000 and standard deviation 800.
■
EXERCISES Section 3.3 (29–45)
29. The pmf of the amount of memory X (GB) in a purchased
a. Compute E(X), E(X 2 ), and V(X).
flash drive was given in Example 3.13 as
b. If the price of a freezer having capacity X cubic feet is , what is the expected price paid by the next
x
1 2 4 8 16 customer to buy a freezer?
c. What is the variance of the price 25X 2 8.5 paid by the
next customer?
d. Suppose that although the rated capacity of a freezer is
Compute the following:
X, the actual capacity is h(X) 5 X 2 .01X 2 . What is the
a. E(X)
expected actual capacity of the freezer purchased by the
b. V(X) directly from the definition
next customer?
c. The standard deviation of X d. V(X) using the shortcut formula
33. Let X be a Bernoulli rv with pmf as in Example 3.18.
a. Compute E(X 2 ).
30. An individual who has automobile insurance from a certain
b. Show that V(X) 5 p(1 2 p) .
company is randomly selected. Let Y be the number of mov-
c. Compute E(X 79 ).
ing violations for which the individual was cited during the last 3 years. The pmf of Y is
34. Suppose that the number of plants of a particular type found in a rectangular sampling region (called a quadrat by ecolo-
y
0 1 2 3 gists) in a certain geographic area is an rv X with pmf
p(y) 3 .60 .25 .10 .05 cx x 5 1, 2, 3, . . .
p(x) 5 e
0 otherwise
a. Compute E(Y).
Is E(X) finite? Justify your answer (this is another distribu-
b. Suppose an individual with Y violations incurs a sur-
charge of 100Y 2 . Calculate the expected amount of the tion that statisticians would call heavy-tailed).
surcharge.
35. A small market orders copies of a certain magazine for its magazine rack each week. Let
X 5 demand for the maga-
31. Refer to Exercise 12 and calculate V(Y) and s Y . Then deter-
zine, with pmf
mine the probability that Y is within 1 standard deviation of its mean value.
x
32. An appliance dealer sells three different models of upright
freezers having 13.5, 15.9, and 19.1 cubic feet of storage
p(x)
space, respectively. Let
X 5 the amount of storage space
purchased by the next customer to buy a freezer. Suppose
Suppose the store owner actually pays 2.00 for each copy of
that X has pmf
the magazine and the price to customers is 4.00. If magazines left at the end of the week have no salvage value, is it better to
x
13.5 15.9 19.1 order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a func-
tion of demand X, and then compute the expected revenue.]
CHAPTER 3 Discrete Random Variables and Probability Distributions
36. Let X be the damage incurred (in ) in a certain type of acci-
40. a. Draw a line graph of the pmf of X in Exercise 35. Then
dent during a given year. Possible X values are 0, 1000,
determine the pmf of 2X and draw its line graph. From
5000, and 10000, with probabilities .8, .1, .08, and .02,
these two pictures, what can you say about V(X) and
respectively. A particular company offers a 500 deductible
V(2X) ?
policy. If the company wishes its expected profit to be 100,
b. Use the proposition involving V(aX 1 b) to establish a
what premium amount should it charge?
general relationship between V(X) and V(2X) .
37. The n candidates for a job have been ranked 1, 2, 3, . . . , n.
41. Use the definition in Expression (3.13) to prove that
Let
X 5 the rank of a randomly selected candidate, so that
V(aX 1 b) 5 a 2 s 2 X . [Hint: With , h(X) 5 aX 1 b
X has pmf
E[h(X)] 5 am 1 b where .] m 5 E(X)
1n x 5 1, 2, 3, . . . , n
42. Suppose and E(X) 5 5 E[X(X 2 1)] 5 27.5 . What is
p(x) 5 e 2 2
0 otherwise
a. E(X )? [Hint: E[X(X 2 1)] 5 E[X
2 X] 5
E(X 2 ) 2 E(X) ]?
(this is called the discrete uniform distribution). Compute
b. V(X)?
E(X) and V(X) using the shortcut formula. [Hint: The sum
c. The general relationship among the quantities E(X),
of the first n positive integers is n(n 1 1)2 , whereas the
E[X(X 2 1)] , and V(X)?
sum of their squares is n(n 1 1)(2n 1 1)6 .]
43. Write a general rule for E(X 2 c) where c is a constant.
38. Let
X 5 the outcome when a fair die is rolled once. If
What happens when you let c5m , the expected value of X?
before the die is rolled you are offered either (13.5) dollars or h(X) 5 1X dollars, would you accept the guaranteed
44. A result called Chebyshev’s inequality states that for any
amount or would you gamble? [Note: It is not generally true
probability distribution of an rv X and any number k that is
P( u X 2 m u ks) 1k that .] 2 1E(X) 5 E(1X) at least 1, . In words, the proba-
bility that the value of X lies at least k standard deviations
39. A chemical supply company currently has in stock 100 lb of
from its mean is at most 1k 2 .
a certain chemical, which it sells to customers in 5-lb
a. What is the value of the upper bound for k52 ? k53
batches. Let
X 5 the number of batches ordered by a ran-
k54 ?? ? k55 k 5 10
domly chosen customer, and suppose that X has pmf
b. Compute m and s for the distribution of Exercise 13. Then evaluate P( u X 2 m u ks) for the values of k
x
1 2 3 4 given in part (a). What does this suggest about the upper
bound relative to the corresponding probability? c. Let X have possible values 21 , 0, and 1, with probabilities
Compute E(X) and V(X). Then compute the expected num-
18 , , and , respectively. What is 9 18 P( u X 2 m u 3s) ,
ber of pounds left after the next customer’s order is shipped
and how does it compare to the corresponding bound?
and the variance of the number of pounds left. [Hint: The
d. Give a distribution for which P( u X 2 m u 5s) 5 .04 .
number of pounds left is a linear function of X.]
45. If , aXb show that a E(X) b .