Chapter 5

Let \(X\) be a random variable with probability density function $$ f(x)=\left\\{\begin{array}{ll} c\left(1-x^{2}\right) & -1

A system consisting of one original unit plus a spare can function for a random amount of time \(X .\) If the density of \(X\) is given (in units of months) by $$ f(x)=\left\\{\begin{array}{ll} C x e^{-x / 2} & x>0 \\ 0 & x \leq 0 \end{array}\right. $$ what is the probability that the system functions for at least 5 months?

Consider the function $$ f(x)=\left\\{\begin{array}{ll} C\left(2 x-x^{3}\right) & 0

The probability density function of \(X\), the lifetime of a certain type of electronic device (measured in hours), is given by $$ f(x)=\left\\{\begin{array}{ll} \frac{10}{x^{2}} & x>10 \\ 0 & x \leq 10 \end{array}\right. $$ (a) \(\operatorname{Find} P\\{X>20\\}\) (b) What is the cumulative distribution function of \(X ?\) (c) What is the probability that of 6 such types of devices, at least 3 will function for at least 15 hours? What assumptions are you making?

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function $$ f(x)=\left\\{\begin{array}{ll} 5(1-x)^{4} & 0

Compute \(E[X]\) if \(X\) has a density function given by (a) \(f(x)=\left\\{\begin{array}{ll}\frac{1}{4} x e^{-x / 2} & x>0 \\ 0 & \text { otherwise }\end{array}\right.\) (b) \(f(x)=\left\\{\begin{array}{ll}c\left(1-x^{2}\right) & -15 \\ 0 & x \leq 5\end{array}\right.\)

The lifetime in hours of an electronic tube is a random variable having a probability density function given by $$ f(x)=x e^{-x} \quad x \geq 0 $$ Compute the expected lifetime of such a tube.

Consider Example 4 b of Chapter \(4,\) but now suppose that the seasonal demand is a continuous random variable having probability density function \(f .\) Show that the optimal amount to stock is the value \(s^{*}\) that satisfies $$ F\left(s^{*}\right)=\frac{b}{b+\ell} $$ where \(b\) is net profit per unit sale, \(\ell\) is the net loss per unit unsold, and \(F\) is the cumulative distribution function of the seasonal demand.

Trains headed for destination \(A\) arrive at the train station at 15 -minute intervals starting at 7 A.M., whereas trains headed for destination \(B\) arrive at 15 -minute intervals starting at 7: 05 A.M. (a) If a certain passenger arrives at the station at a time uniformly distributed between 7 and 8 A.M. and then gets on the first train that arrives, what proportion of time does he or she go to destination \(A ?\) (b) What if the passenger arrives at a time uniformly distributed between 7: 10 and 8: 10 A.M.?

A point is chosen at random on a line segment of length \(L .\) Interpret this statement, and find the probability that the ratio of the shorter to the longer segment is less than \(\frac{1}{4}\).

You arrive at a bus stop at 10 A.M., knowing that the bus will arrive at some time uniformly distributed between 10 and 10: 30 (a) What is the probability that you will have to wait longer than 10 minutes? (b) If, at \(10: 15,\) the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

If \(X\) is a normal random variable with parameters \(\mu=10\) and \(\sigma^{2}=36,\) compute (a) \(P\\{X>5\\}\) (b) \(P\\{416\\}\).

The annual rainfall (in inches) in a certain region is normally distributed with \(\mu=40\) and \(\sigma=4 .\) What is the probability that starting with this year, it will take more than 10 years before a year occurs having a rainfall of more than 50 inches? What assumptions are you making?

The salaries of physicians in a certain speciality are approximately normally distributed. If 25 percent of these physicians earn less than \(\$ 180,000\) and 25 percent earn more than \(\$ 320,000,\) approximately what fraction earn (a) less than \(\$ 200,000 ?\) (b) between \(\$ 280,000\) and \(\$ 320,000 ?\)

Suppose that \(X\) is a normal random variable with mean \(5 .\) If \(P\\{X>9\\}=.2,\) approximately what is \(\operatorname{Var}(X) ?\)

Let \(X\) be a normal random variable with mean 12 and variance \(4 .\) Find the value of \(c\) such that \(P\\{X>c\\}=.10\).

If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain (a) at least 50 who are in favor of the proposition; (b) between 60 and 70 inclusive who are in favor; (c) fewer than 75 in favor.

Suppose that the height, in inches, of a 25 -year-old man is a normal random variable with parameters \(\mu=71\) and \(\sigma^{2}=6.25 .\) What percentage of 25 -year-old men are taller than 6 feet, 2 inches? What percentage of men in the 6-footer club are taller than 6 feet, 5 inches?

Every day Jo practices her tennis serve by continually serving until she has had a total of 50 successful serves. If each of her serves is, independently of previous ones, successful with probability \(.4,\) approximately what is the probability that she will need more than 100 serves to accomplish her goal? Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly 100 times. What must be true about her first 100 serves if she is to reach her goal?

One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively. If the number 6 appears exactly 200 times, find the probability that the number 5 will appear less than 150 times.

The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are normally distributed with parameters \(\mu=1.4 \times 10^{6}\) hours and \(\sigma=\) \(3 \times 10^{5}\) hours. What is the approximate probability that a batch of 100 chips will contain at least 20 whose lifetimes are less than \(1.8 \times 10^{6} ?\)

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability .95 Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. If the coin lands on heads 525 or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than 525 times. then we shall conclude that it is a fair coin. If the coin is actually fair, what is the probability that we shall reach a false conclusion? What would it be if the coin were biased?

In 10,000 independent tosses of a coin, the coin landed on heads 5800 times. Is it reasonable to assume that the coin is not fair? Explain.

Twelve percent of the population is left handed. Approximate the probability that there are at least 20 lefthanders in a school of 200 students. State your assumptions.

A model for the movement of a stock supposes that if the present price of the stock is \(s,\) then after one period, it will be either \(u s\) with probability \(p\) or \(d s\) with probability \(1-p .\) Assuming that successive movements are independent, approximate the probability that the stock's price will be up at least 30 percent after the next 1000 periods if \(u=1.012, d=0.990,\) and \(p=.52\).

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with \(\mu=4\) and \(\sigma^{2}=4,\) whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters \((6,9) .\) A point is randomly chosen on the image and has a reading of \(5 .\) If the fraction of the image that is black is \(\alpha,\) for what value of \(\alpha\) would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?

(a) \(A\) fire station is to be located along a road of length \(A, A<\infty\). If fires occur at points uniformly chosen on \((0, A),\) where should the station be located so as to minimize the expected distance from the fire? That is, choose \(a\) so as to $$ \operatorname{minimize} E[|X-a|] $$ when \(X\) is uniformly distributed over \((0, A)\).

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter \(\lambda=\frac{1}{2} .\) What is (a) the probability that a repair time exceeds 2 hours? (b) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours?

The number of years a radio functions is exponentially distributed with parameter \(\lambda=\frac{1}{8} .\) If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter \(\frac{1}{20}\). 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).

The lung cancer hazard rate \(\lambda(t)\) of a \(t\) -year-old male smoker is such that $$ \lambda(t)=.027+.00025(t-40)^{2} \quad t \geq 40 $$ Assuming that a 40 -year-old male smoker survives all other hazards, what is the probability that he survives to (a) age 50 and (b) age 60 without contracting lung cancer?

Suppose that the life distribution of an item has the hazard rate function \(\lambda(t)=t^{3}, t>0 .\) What is the probability that (a) the item survives to age \(2 ?\) (b) the item's lifetime is between .4 and \(1.4 ?\) (c) a 1-year-old item will survive to age \(2 ?\)

5.37. If \(X\) is uniformly distributed over \((-1,1),\) find (a) \(P\left\\{|X|>\frac{1}{2}\right\\}\) (b) the density function of the random variable \(|X| .\)

Let \(Y\) be a lognormal random variable (see Example 7e for its definition) and let \(c>0\) be a constant. Answer true or false to the following, and then give an explanation for your answer. (a) \(c Y\) is lognormal; (b) \(c+Y\) is lognormal.