Chapter 6

A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by \(N_{1}\) the number of tests made until the first defective is identified and by \(N_{2}\) the number of additional tests until the second defective is identified. Find the joint probability mass function of \(N_{1}\) and \(N_{2}.\)

Consider a sequence of independent Bernoulli trials, each of which is a success with probability \(p .\) Let \(X_{1}\) be the number of failures preceding the first success, and let \(X_{2}\) be the number of failures between the first two successes. Find the joint mass function of \(X_{1}\) and \(X_{2}.\)

The joint probability density function of \(X\) and \(Y\) is given by $$f(x, y)=c\left(y^{2}-x^{2}\right) e^{-y} \quad-y \leq x \leq y, 0< y<\infty$$ (a) Find \(c\). (b) Find the marginal densities of \(X\) and \(Y\). (c) Find \(E[X]\).

The joint probability density function of \(X\) and \(Y\) is given by $$f(x, y)=\frac{6}{7}\left(x^{2}+\frac{x y}{2}\right) \quad 0< x<1,0< y<2$$ (a) Verify that this is indeed a joint density function. (b) Compute the density function of \(X\). (c) Find \(P\\{X>Y\\}\). (d) Find \(P\left\\{Y>\frac{1}{2} | X<\frac{1}{2}\right\\}\). (e) Find \(E[X]\). (f) Find \(E[Y]\).

The joint probability density function of \(X\) and \(Y\) is given by $$f(x, y)=e^{-(x+y)} \quad 0 \leq x<\infty, 0 \leq y<\infty$$ Find (a) \(P\\{X< Y\\}\) and (b) \(P\\{X< a\\}\)

A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 15 percent will purchase a plasma television set, and 40 percent will just be browsing. If 5 customers enter his store on a given day, what is the probability that he will sell exactly 2 ordinary sets and 1 plasma set on that day?

The number of people who enter a drugstore in a given hour is a Poisson random variable with parameter \(\lambda=10 .\) Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?

An ambulance travels back and forth at a constant speed along a road of length \(L\). At a certain moment of time, an accident occurs at a point uniformly distributed on the road. [That is, the distance of the point from one of the fixed ends of the road is uniformly distributed over \((0, L) .]\) Assuming that the ambulance's location at the moment of the accident is also uniformly distributed, and assuming independence of the variables, compute the distribution of the distance of the ambulance from the accident.

The random vector \((X, Y)\) is said to be uniformly distributed over a region \(R\) in the plane if, for some constant \(c,\) its joint density is $$f(x, y)=\left\\{\begin{array}{ll}c & \text { if }(x, y) \in R \\\0 & \text { otherwise }\end{array}\right.$$ (a) Show that \(1 / c=\) area of region \(R\) Suppose that \((X, Y)\) is uniformly distributed over the square centered at (0,0) and with sides of length 2. (b) Show that \(X\) and \(Y\) are independent, with each being distributed uniformly over (-1,1). (c) What is the probability that \((X, Y)\) lies in the circle of radius 1 centered at the origin? That is, find \(P\left\\{X^{2}+Y^{2} \leq 1\right\\}.\)

Three points \(X_{1}, X_{2}, X_{3}\) are selected at random on a line \(L .\) What is the probability that \(X_{2}\) lies between \(X_{1}\) and \(X_{3} ?\)

Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points \(X\) and \(Y\) are independent random variables such that \(X\) is uniformly distributed over \((0, L / 2) \text { and } Y \text { is uniformly distributed over }(L / 2, L) .]\) Find the probability that the distance between the two points is greater than \(L / 3 .\)

Show that \(f(x, y)=1 / x, 0

Let $$f(x, y)=24 x y \quad 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq x+y \leq 1$$ and let it equal 0 otherwise. (a) Show that \(f(x, y)\) is a joint probability density function. (b) Find \(E[X]\). (c) Find \(E[Y]\).

The joint density function of \(X\) and \(Y\) is $$f(x, y)=\left\\{\begin{array}{ll}x+y & 0< x<1,0< y<1 \\ 0 & \text { otherwise }\end{array}\right.$$ (a) Are \(X\) and \(Y\) independent? (b) Find the density function of \(X\). (c) Find \(P\\{X+Y<1\\}\).

The random variables \(X\) and \(Y\) have joint density function $$f(x, y)=12 x y(1-x) \quad 0

Consider independent trials, each of which results in outcome \(i, i=0,1, \ldots, k,\) with probability \(p_{i}, \sum_{i=0}^{k} p_{i}=1.\) Let \(N\) denote the number of trials needed to obtain an outcome that is not equal to \(0,\) and let \(X\) be that outcome. (a) Find \(P\\{N=n\\}, n \geq 1\). (b) Find \(P\\{X=j\\}, j=1, \ldots, k\). (c) Show that \(P\\{N=n, X=j\\}=P\\{N=n\\} P\\{X=j\\}\). (d) Is it intuitive to you that \(N\) is independent of \(X ?\) (e) Is it intuitive to you that \(X\) is independent of \(N ?\)

Suppose that \(10^{6}\) people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over \(\left(0,10^{6}\right) .\) Let \(N\) denote the number that arrive in the first hour. Find an approximation for \(P\\{N=i\\}.\)

If \(X_{1}\) and \(X_{2}\) are independent exponential random variables with respective parameters \(\lambda_{1}\) and \(\lambda_{2},\) find the distribution of \(Z=X_{1} / X_{2} .\) Also compute \(P\left\\{X_{1}

The time that it takes to service a car is an exponential random variable with rate 1. (a) If A. J. brings his car in at time 0 and M. J. brings her car in at time \(t,\) what is the probability that M. J.'s car is ready before A. J's car? (Assume that service times are independent and service begins upon arrival of the car. (b) If both cars are brought in at time \(0,\) with work starting on M. J's car only when A. J.'s car has been completely serviced, what is the probability that M. J's car is ready before time \(2 ?\)

The gross weekly sales at a certain restaurant are a normal random variable with mean \(\$ 2200\) and standard deviation \(\$ 230 .\) What is the probability that (a) the total gross sales over the next 2 weeks exceeds \(\$ 5000\) (b) weekly sales exceed \(\$ 2000\) in at least 2 of the next 3 weeks? What independence assumptions have you made?

Jill's bowling scores are approximately normally distributed with mean 170 and standard deviation \(20,\) while Jack's scores are approximately normally distributed with mean 160 and standard deviation \(15 .\) If Jack and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that (a) Jack's score is higher; (b) the total of their scores is above \(350 .\)

According to the U.S. National Center for Health Statistics, 25.2 percent of males and 23.6 percent of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that (a) at least 110 of these 400 people never eat breakfast; (b) the number of the women who never eat breakfast is at least as large as the number of the men who never eat breakfast.

Monthly sales are independent normal random variables with mean 100 and standard deviation 5. (a) Find the probability that exactly 3 of the next 6 months have sales greater than \(100 .\) (b) Find the probability that the total of the sales in the next 4 months is greater than \(420 .\)

The expected number of typographical errors on a page of a certain magazine is .2. What is the probability that an article of 10 pages contains (a) 0 and (b) 2 or more typographical errors? Explain your reasoning!

The monthly worldwide average number of airplane crashes of commercial airlines is \(2.2 .\) What is the probability that there will be (a) more than 2 such accidents in the next month? (b) more than 4 such accidents in the next 2 months? (c) more than 5 such accidents in the next 3 months? Explain your reasoning!

In Problem \(6.3,\) calculate the conditional probability mass function of \(Y_{1}\) given that (a) \(Y_{2}=1\) (b) \(Y_{2}=0\)

Choose a number \(X\) at random from the set of numbers \(\\{1,2,3,4,5\\} .\) Now choose a number at random from the subset no larger than \(X,\) that is, from \(\\{1, \ldots, X\\} .\) Call this second number \(Y\). (a) Find the joint mass function of \(X\) and \(Y\). (b) Find the conditional mass function of \(X\) given that \(Y=i .\) Do it for \(i=1,2,3,4,5\). (c) Are \(X\) and \(Y\) independent? Why?

Two dice are rolled. Let \(X\) and \(Y\) denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of \(Y\) given \(X=i,\) for \(i=\) \(1,2, \ldots, 6 .\) Are \(X\) and \(Y\) independent? Why?

The joint probability mass function of \(X\) and \(Y\) is given by $$\begin{array}{ll}p(1,1)=\frac{1}{8} & p(1,2)=\frac{1}{4} \\\p(2,1)=\frac{1}{8} & p(2,2)=\frac{1}{2}\end{array}$$ (a) Compute the conditional mass function of \(X\) given \(Y=i, i=1,2\). (b) Are \(X\) and \(Y\) independent? (c) Compute \(P\\{X Y \leq 3\\}, P\\{X+Y>2\\}, P\\{X / Y>1\\}\).

The joint density function of \(X\) and \(Y\) is given by $$f(x, y)=x e^{-x(y+1)} \quad x>0, y>0$$ (a) Find the conditional density of \(X,\) given \(Y=y,\) and that of \(Y,\) given \(X=x\). (b) Find the density function of \(Z=X Y\).

An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is \(\lambda\) is Poisson distributed with mean \(\lambda .\) They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameters \(s\) and \(\alpha .\) If a newly insured person has \(n\) accidents in her first year, find the conditional density of her accident parameter. Also, determine the expected number of accidents that she will have in the following year.

If \(X_{1}, X_{2}, X_{3}\) are independent random variables that are uniformly distributed over \((0,1),\) compute the probability that the largest of the three is greater than the sum of the other two.

If 3 trucks break down at points randomly distributed on a road of length \(L,\) find the probability that no 2 of the trucks are within a distance \(d\) of each other when \(d \leq L / 2.\)

Consider a sample of size 5 from a uniform distribution over \((0,1) .\) Compute the probability that the median is in the interval \(\left(\frac{1}{4}, \frac{3}{4}\right).\)

Let \(X_{(1)}, X_{(2)}, \ldots, X_{(n)}\) be the order statistics of a set of \(n\) independent uniform (0,1) random variables. Find the conditional distribution of \(X_{(n)}\) given that \(X_{(1)}=s_{1}, X_{(2)}=s_{2}, \dots, X_{(n-1)}=s_{n-1}\)

Let \(Z_{1}\) and \(Z_{2}\) be independent standard normal random variables. Show that \(X, Y\) has a bivariate normal distribution when \(X=Z_{1}, Y=Z_{1}+Z_{2}.\)

Derive the distribution of the range of a sample of size 2 from a distribution having density function \(f(x)=\) \(2 x, 0

Let \(X\) and \(Y\) denote the coordinates of a point uniformly chosen in the circle of radius 1 centered at the origin. That is, their joint density is $$f(x, y)=\frac{1}{\pi} \quad x^{2}+y^{2} \leq 1$$ Find the joint density function of the polar coordinates \(R=\left(X^{2}+Y^{2}\right)^{1 / 2}\) and \(\Theta=\tan ^{-1} Y / X\)

If \(X\) and \(Y\) are independent random variables both uniformly distributed over \((0,1),\) find the joint density function of \(R=\sqrt{X^{2}+Y^{2}}, \Theta=\tan ^{-1} Y / X.\)

\(X\) and \(Y\) have joint density function $$f(x, y)=\frac{1}{x^{2} y^{2}} \quad x \geq 1, y \geq 1$$ (a) Compute the joint density function of \(U=X Y, V=\) \(X / Y.\) (b) What are the marginal densities?

Repeat Problem 6.56 when \(X\) and \(Y\) are independent exponential random variables, each with parameter \(\lambda=1.\)

If \(X_{1}\) and \(X_{2}\) are independent exponential random variables, each having parameter \(\lambda,\) find the joint density function of \(Y_{1}=X_{1}+X_{2}\) and \(Y_{2}=e^{X_{1}}.\)

If \(X, Y,\) and \(Z\) are independent random variables having identical density functions \(f(x)=e^{-x}, 0

Consider an urn containing \(n\) balls numbered \(1, \ldots, n,\) and suppose that \(k\) of them are randomly withdrawn. Let \(X_{i}\) equal 1 if ball number \(i\) is removed and let \(X_{i}\) be 0 otherwise. Show that \(X_{1}, \ldots, X_{n}\) are exchangeable.