Introduction to Probability and Statistics for Engineers and Scientists

Problems 199

35.The height of adult women in the United States is normally distributed with

mean 64.5 inches and standard deviation 2.4 inches. Find the probability that

a randomly chosen woman is

(a)less than 63 inches tall;

(b) less than 70 inches tall;

(c) between 63 and 70 inches tall.

(d) Alice is 72 inches tall. What percentage of women is shorter than Alice?

(e)Find the probability that the average of the heights of two randomly chosen

women exceeds 66 inches.

(f ) Repeat part (e) for four randomly chosen women.

36.An IQ test produces scores that are normally distributed with mean value 100

and standard deviation 14.2. The top 1 percent of all scores are in what range?

37.The time (in hours) required to repair a machine is an exponentially distributed

random variable with parameterλ=1.

(a)What is the probability that a repair time exceeds 2 hours?

(b) What is the conditional probability that a repair takes at least 3 hours, given

that its duration exceeds 2 hours?

38.The number of years a radio functions is exponentially distributed with parameter

λ=^18. If Jones buys a used radio, what is the probability that it will be working

after an additional 10 years?

39.Jones figures that the total number of thousands of miles that a used auto can be

driven before it would need to be junked is an exponential random variable with

parameter 201. Smith has a used car that he claims has been driven only 10,000

miles. If Jones purchases the car, what is the probability that she would get at least

20,000 additional miles out of it? Repeat under the assumption that the lifetime

mileage of the car is not exponentially distributed but rather is (in thousands of

miles) uniformly distributed over (0, 40).

*40.LetX 1 ,X 2 ,...,Xndenote the firstninterarrival times of a Poisson process and

setSn=

∑n

i= 1 Xi.

(a)What is the interpretation ofSn?

(b) Argue that the two events{Sn≤t}and{N(t)≥n}are identical.

(c) Use part (b) to show that

P{Sn≤t}= 1 −

∑n−^1

j= 0

e−λt(λt)j/j!

(d) By differentiating the distribution function ofSngiven in part (c), conclude

thatSnis a gamma random variable with parametersnandλ. (This result

also follows from Corollary 5.7.2.)

* From optional sections.