SOLUTIONS TO CHAPTER 5 HOMEWORK
Exercise 10 (Additional). Suppose that the length of long distance phone calls,
1
on the
measured in minutes, is known to be distributed with density f (x) = 3x
interval [1, b) for the appropriate value of b. Find b and then answer the following
questions.
Rb
We know that we must have 1 = 1 31 x1 dx = 13 ln(x) |b1 = 13 ln(b). Thus, b = e3 .
a. X = the length in minutes of a phone call
b. Is X continuous or discrete? continuous
1
c. X ∼ with density f (x) = 3x
3
R
3
e
1
d. µ = e 3−1 ≈ 6.36, from 1 x 3x
dx
q
R e3 2 1
R e3
6
e6 −1
2
≈ 8.19 from σ = 1 x 3x dx = 31 1 xdx = e 6−1
e. σ =
6
f. Draw a graph of the probability distribution. Label the axes. Solution not
provided here.
g. Find the probability that a phone call lasts less than 9 minutes.
It will be
to find the cumulative distribution function
R xhelpful
1
F (x) = 1 3t
dt = 31 ln(x).
h. Find the probability that a phone call lasts more than 9 minutes.
P (X > 9) = 1 − F (9) = 1 − 13 ln(9).
i. Find the probability that a phone call lasts between 7 and 9 minutes.
P (7 < X < 9) = F (9) − F (7) = 31 (ln(9) − ln(7)).
j. If 25 phone calls are made one after another, on average, what would you
expect the total to be? Why?
3
The total should be about 25µ = 25
3 (e − 1).
1