Exercise 5.11.1¶
Let X and P be random variables. Suppose that the conditional distribution of X given P = p is the binomial distribution with parameters n and p. Suppose that the distribution of P is the beta distribution with parameters α = 1 and β = 1. Find the marginal distribution of X.
Suppose that X, Y , and Z are i.i.d. random variables and each has the standard normal distribution. Evaluate Pr(3X + 2Y <6Z − 7).
Suppose thatX and Y are independent Poisson random variables such thatVar(X) + Var(Y ) = 5. Evaluate Pr(X + Y <2).
Suppose that X has a normal distribution such that Pr(X < 116) = 0.20 and Pr(X < 328) = 0.90. Determine the mean and the variance of X.
Suppose that a random sample of four observations is drawn from the Poisson distribution with mean λ, and let X denote the sample mean. Show that Pr
X< 1 2 = (4λ + 1)e −4λ. 6. The lifetime X of an electronic component has the exponential distribution such that Pr(X ≤ 1000) = 0.75. What is the expected lifetime of the component? 7. Suppose that X has the normal distribution with mean μ and variance σ2. Express E(X3) in terms of μ and σ2. 8. Suppose that a random sample of 16 observations is drawn from the normal distribution with mean μ and standard deviation 12, and that independently another random sample of 25 observations is drawn from the normal distribution with the same mean μ and standard deviation 20. Let X and Y denote the sample means of the two samples. Evaluate Pr(|X − Y | < 5). 9. Suppose that men arrive at a ticket counter according to a Poisson process at the rate of 120 per hour, andwomen arrive according to an independent Poisson process at the rate of 60 per hour. Determine the probability that four or fewer people arrive in a one-minute period. 10. Suppose that X1, X2, . . . are i.i.d. random variables, each of which has m.g.f. ψ(t). Let Y = X1 + . . . + XN, where the number of terms N in this sum is a random variable having the Poisson distribution with mean λ. Assume thatN andX1, X2, . . . are independent, and Y = 0 if N = 0. Determine the m.g.f. of Y . 11. Every Sunday morning, two children, Craig and Jill, independently try to launch their model airplanes. On each Sunday, Craig has probability 1/3 of a successful launch, and Jill has probability 1/5 of a successful launch. Determine the expected number of Sundays required until at least one of the two children has a successful launch. 12. Suppose that a fair coin is tossed until at least one head and at least one tail have been obtained. Let X denote the number of tosses that are required. Find the p.f. of X. 13. Suppose that a pair of balanced dice are rolled 120 times, and let X denote the number of rolls on which the sum of the two numbers is 12. Use the Poisson approximation to approximate Pr(X = 3). 14. Suppose that X1, . . . , Xn form a random sample from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1, . . . , Xn }, Yn = max{X1, . . . , Xn }, and W = Yn − Y1. Show that each of the random variables Y1, Yn, and W has a beta distribution. 15. Suppose that events occur in accordance with a Poisson process at the rate of five events per hour. a. Determine the distribution of the waiting time T1 until the first event occurs. b. Determine the distribution of the total waiting time Tk until k events have occurred. c. Determine the probability that none of the first k events will occur within 20 minutes of one another. 16. Suppose that five components are functioning simultaneously, that the lifetimes of the components are i.i.d., and that each lifetime has the exponential distribution with parameter β. Let T1 denote the time from the beginning of the process until one of the components fails; and let T5 denote the total time until all five components have failed. Evaluate Cov(T1, T5). 17. Suppose thatX1 andX2 are independent random variables, andXi has the exponential distribution with parameter βi (i = 1, 2). Show that for each constant k > 0, Pr(X1 > kX2) = β2 kβ1 + β2 . 18. Suppose that 15,000 people in a city with a population of 500,000 are watching a certain television program. If 200 people in the city are contacted at random, what is the approximate probability that fewer than four of them are watching the program? 19. Suppose that it is desired to estimate the proportion of persons in a large population who have a certain characteristic. A random sample of 100 persons is selected from the population without replacement, and the proportion X of persons in the sample who have the characteristic is observed. Show that, no matter how large the population is, the standard deviation of X is at most 0.05. 20. Suppose that X has the binomial distribution with parameters n and p, and that Y has the negative binomial distribution with parameters r and p, where r is a positive integer. Show that Pr(X < r) = Pr(Y > n − r) by showing 346 Chapter 5 Special Distributions that both the left side and the right side of this equation can be regarded as the probability of the same event in a sequence of Bernoulli trials with probability p of success. 21. Suppose thatXhas the Poisson distribution with mean λt, and that Y has the gamma distribution with parameters α = k and β = λ, where k is a positive integer. Show that Pr(X ≥ k) = Pr(Y ≤ t) by showing that both the left side and the right side of this equation can be regarded as the probability of the same event in a Poisson process in which the expected number of occurrences per unit of time is λ. 22. Suppose that X is a random variable having a continuous distribution with p.d.f. f (x) and c.d.f. F(x), and for which Pr(X > 0) = 1. Let the failure rate h(x) be as defined in Exercise 18 of Sec. 5.7. Show that exp − x 0 h(t) dt = 1− F(x). 23. Suppose that 40 percent of the students in a large population are freshmen, 30 percent are sophomores, 20 percent are juniors, and 10 percent are seniors. Suppose that 10 students are selected at random from the population, and let X1, X2, X3, X4 denote, respectively, the numbers of freshmen, sophomores, juniors, and seniors that are obtained. a. Determine ρ(Xi, Xj ) for each pair of values i and j (i < j ). b. For what values of i and j (i<j) is ρ(Xi, Xj ) most negative? c. For what values of i and j (i<j) is ρ(Xi, Xj ) closest to 0? 24. Suppose that X1 and X2 have the bivariate normal distribution with means μ1 and μ2, variances σ2 1 and σ2 2, and correlation ρ. Determine the distribution ofX1− 3X2. 25. Suppose that X has the standard normal distribution, and the conditional distribution of Y givenX is the normal distribution with mean 2X − 3and variance 12. Determine the marginal distribution of Y and the value of ρ(X, Y). 26. Suppose that and have a bivariate normal distribution with . Evaluate .