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A First Course In Probability 9th Edition Sheldon Ross - Solutions

In Problem 5, calculate the conditional probability mass function of Y1 given that(a) Y2 = 1;(b) Y2 = 0.Problem 5Repeat Problem 3a when the ball selected is replaced in the urn before the next selection.Problem 3In Problem 2, suppose that the white balls are numbered, and let Yi equal 1 if the ith
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, . . . ,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
Repeat Problem 2 when the ball selected is replaced in the urn before the next selection.Problem 2Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let Xi equal 1 if the ith ball selected is white, and let it equal 0 otherwise. Give the joint
The joint density function of X and Y is given by f (x, y) = xe−x(y+1)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 = XY.
The joint density of X and Y isf (x, y) = c(x2 − y2)e−x 0 ≤ x < ∞, −x ≤ y ≤ xFind the conditional distribution of Y, given X = x.
An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is λ is Poisson distributed with mean λ. They also suppose that the parameter value of a newly insured person can be assumed to be the value of a
If X1, X2, X3 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.
A complex machine is able to operate effectively as long as at least 3 of its 5 motors are functioning. If each motor independently functions for a random amount of time with density function f (x) = xe−x, x > 0, compute the density function of the length of time that the machine functions.
If X1, X2, X3, X4, X5 are independent and identically distributed exponential random variables with the parameter λ, compute (a) P{min(X1, . . . ,X5) ≤ a}; (b) P{max(X1, . . . ,X5) ≤ a}.
Repeat Problem 3a when the ball selected is replaced in the urn before the next selection. Problem 3 In Problem 2, suppose that the white balls are numbered, and let Yi equal 1 if the ith white ball is selected and 0 otherwise. Find the joint probability mass function of (a) Y1, Y2; Problem
Let Z1 and Z2 be independent standard normal random variables. Show that X, Y has a bivariate normal distribution when X = Z1, Y = Z1 + Z2.
Derive the distribution of the range of a sample of size 2 from a distribution having density functionf(x) = 2x, 0 < x < 1.
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 isf(x, y) = 1/πx2 + y2 ≤ 1Find the joint density function of the polar coordinatesR = (X2 + Y2)1/2 and Θ = tan−1 Y/X.
If U is uniform on (0, 2π) and Z, independent of U, is exponential with rate 1, show directly (without using the results of Example 7b) that X and Y defined byX = √2ZcosUY = √2ZsinUare independent standard normal random variables.
X and Y have joint density functionf(x, y) = 1/x2y2 x ≥ 1, y ≥ 1(a) Compute the joint density function of U = XY, V = X/Y.(b) What are the marginal densities?
If X and Y are independent and identically distributed uniform random variables on (0, 1), compute the joint density ofU = X + Y, V = X/Y
If X1 and X2 are independent exponential random variables, each having parameter λ, find the joint density function of Y1 = X1 + X2 and Y2 = eX1.
If X, Y, and Z are independent random variables having identical density functions f(x) = e−x, 0 < x < ∞, derive the joint distribution of U = X + Y, V = X + Z, W = Y + Z.
In Example 8b, letShow that Y1, . . . ,Yk, Yk+1 are exchangeable. Yk+1 is the number of balls one must observe to obtain a special ball if one considers the balls in their reverse order of withdrawal.
Consider an urn containing n balls numbered 1, . . . , n, and suppose that k of them are randomly withdrawn. Let Xi equal 1 if ball number i is removed and let Xi be 0 otherwise. Show that X1, . . . ,Xn are exchangeable.
The joint probability density function of X and Y is given by f (x, y) = 6/7(x2 + xy/2) 0 < x < 1, 0 < y < 2 (a) Compute the density function of X. (b) Find P{X > Y}. (c) Find P{Y > 1/2|X < 1/2}.
Verify Equation (1.2).
The lifetimes of batteries are independent exponential random variables, each having parameter λ. A flashlight needs 2 batteries to work. If one has a flashlight and a stockpile of n batteries, what is the distribution of time that the flashlight can operate?
Show that the jointly continuous (discrete) random variables X1, . . . ,Xn are independent if and only if their joint probability density (mass) function f (x1, . . . , xn) can be written asfor nonnegative functions gi(x), i = 1, . . . , n.
In Example 5c we computed the conditional density of a success probability for a sequence of trials when the first n + m trials resulted in n successes. Would the conditional density change if we specified which n of these trials resulted in successes?
Suppose that X and Y are independent geometric random variables with the same parameter p.Without any computations, what do you think is the value ofP{X = i|X + Y = n}?Imagine that you continually flip a coin having probability p of coming up heads. If the second head occurs on the nth flip, what
Consider a sequence of independent trials, with each trial being a success with probability p. Given that the kth success occurs on trial n, show that all possible outcomes of the first n − 1 trials that consist of k − 1 successes and n − k failures are equally likely.
If X and Y are independent binomial random variables with identical parameters n and p, show analytically that the conditional distribution of X given that X + Y = m is the hypergeometric distribution. Also, give a second argument that yields the same result without any computations. Suppose that
Suppose that Xi, i = 1, 2, 3 are independent Poisson random variables with respective means λi, i = 1, 2, 3. Let X = X1 + X2 and Y = X2 + X3. The random vector X, Y is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find P{X = n, Y = m}.
Suppose X and Y are both integer-valued random variables. Letp(i|j) = P(X = i|Y = j)andq(j|i) = P(Y = j|X = i)Show that
Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute (a) P{X1 > X2|X1 > X3}; (b) P{X1 > X2|X1 < X3}; (c) P{X1 > X2|X2 > X3}; (d) P{X1 > X2|X2 < X3}.
Suppose that the number of events occurring in a given time period is a Poisson random variable with parameter λ. If each event is classified as a type i event with probability pi, i = 1, . . . , n, ∑ pi = 1, independently of other events, show that the numbers of type i events that occur, i =
Let U denote a random variable uniformly distributed over (0, 1). Compute the conditional distribution of U given that(a) U > a;(b) U < a;where 0 < a < 1.
Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters (t, β). That is, its density is f(w) = βe−βw(βw)t−1 / Γ(t), w > 0. Suppose also that given that W = w, the number of accidents during that day—call it N—has a Poisson
Let W be a gamma random variable with parameters (t, Î²), and suppose that conditional on W = w, X1, X2, . . . ,Xn are independent exponential random variables with rate w. Show that the conditional distribution of W given that X1 = x1, X2 = x2, . . . ,Xn = xn is gamma with parameters
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the arraythe number 1 in the first row, first column is a saddlepoint. The
If X is exponential with rate λ, find P{[X] = n,X − [X] ≤ x}, where [x] is defined as the largest integer less than or equal to x. Can you conclude that [X] and X − [X] are independent?
Suppose that F(x) is a cumulative distribution function. Show that (a) Fn(x) and (b) 1 − [1 − F(x)]n are also cumulative distribution functions when n is a positive integer.Let X1, . . . ,Xn be independent random variables having the common distribution function F. Define random variables Y and
Show that if n people are distributed at random along a road L miles long, then the probability that no 2 people are less than a distance D miles apart is when D ≤ L/(n − 1), [1 − (n − 1)D/L]n. What if D > L/(n − 1)?
Establish Equation (6.2) by differentiating Equation (6.4).
Verify Equation (6.6), which gives the joint density of X(i) and X(j).
Suggest a procedure for using Buffon’s needle problem to estimate π. Surprisingly enough, this was once a common method of evaluating π.
Let X(1) ≤ X(2) ≤ . . . ≤ X(n) be the ordered values of n independent uniform (0, 1) random variables.Prove that for 1 ≤ k ≤ n + 1,P{X(k) − X(k−1) > t} = (1 − t)nwhere X(0) ≡ 0, X(n+1) ≡ t.
Let X1, . . . ,Xn be a set of independent and identically distributed continuous random variables having distribution function F, and let X(i), i = 1, . . . , n denote their ordered values. If X, independent of the Xi, i = 1, . . . , n, also has distribution F, determine(a) P{X > X(n)};(b) P{X
If X and Y are independent standard normal random variables, determine the joint density function ofU = XV= X/YThen use your result to show that X/Y has a Cauchy distribution.
If X and Y are independent continuous positive random variables, express the density function of (a) Z = X/Y and (b) Z = XY in terms of the density functions of X and Y. Evaluate the density functions in the special case where X and Y are both exponential random variables.
If X and Y are jointly continuous with joint density function fX,Y(x, y), show that X + Y is continuous with density function
(a) If X has a gamma distribution with parameters (t, Î»), what is the distribution of cX, c > 0?(b) Show thathas a gamma distribution with parameters n, Î» when n is a positive integer and Ï‡22n is a chi-squared random variable with 2n degrees of freedom.
Let X and Y be independent continuous random variables with respective hazard rate functions λX(t) and λY(t), and set W = min(X,Y).(a) Determine the distribution function of W in terms of those of X and Y.(b) Show that λW(t), the hazard rate function of W, is given by λW(t) = λX(t) + λY(t)
Let X1, . . . ,Xn be independent exponential random variables having a common parameter λ. Determine the distribution of min(X1, . . . ,Xn).
Consider n independent flips of a coin having probability p of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if n = 5 and the outcome is HHTHT, then there are 3 changeovers. Find the expected number of changeovers.Express the
For a standard normal random variable Z, let Î¼n = E[Zn]. Show thatStart by expanding the moment generating function of Z into a Taylor series about 0 to obtain
Let X be a normal random variable with mean Î¼ and variance Ïƒ2. Use the results of Theoretical Exercise 46 to show thatIn the preceding equation, [n/2] is the largest integer less than or equal to n/2. Check your answer by letting n = 1 and n = 2.
The positive random variable X is said to be a lognormal random variable with parameters μ and σ2 if log(X) is a normal random variable with mean μ and variance σ2. Use the normal moment generating function to find the mean and variance of a lognormal random variable.
Let A1, A2, . . . ,An be arbitrary events, and define Ck = {at least k of the Ai occur}. Show thatLet X denote the number of the Ai that occur. Show that both sides of the preceding equation are equal to E[X].
Let X have moment generating function M(t), and define ψ(t) = logM(t). Show thatψ′′(t)|t=0 = Var(X)
Show how to compute Cov(X, Y) from the joint moment generating function of X and Y.
Suppose that Y is a normal random variable with mean μ and variance σ2, and suppose also that the conditional distribution of X, given that Y = y, is normal with mean y and variance 1.(a) Argue that the joint distribution of X, Y is the same as that of Y + Z, Y when Z is a standard normal random
In the text, we noted thatwhen the Xi are all nonnegative random variables. Since an integral is a limit of sums, one might expect thatwhenever X(t), 0 ‰¤ t Define, for each nonnegative t, the random variable X(t) byNow relate
We say that X is stochastically larger than Y, written X ‰¥st Y, if, for all t.P{X > t} ‰¥ P{Y > t}Show that if X ‰¥st Y, then E[X] ‰¥ E[Y] when(a) X and Y are nonnegative random variables;(b) X and Y are arbitrary random variables.Write X asX = X+ ˆ’ Xˆ’WhereSimilarly,
Show that X is stochastically larger than Y if and only ifE[f(X)] ≥ E[f(Y)]for all increasing functions f .Show that X ≥st Y, then E[f(X)] ≥ E[f(Y)] by showing that f(X) ≥st f(Y) and then using Theoretical Exercise 7.7. To show that if E[f(X)] ≥ E[f(Y)] for all increasing functions f,
A group of n men and n women is lined up at random. (a) Find the expected number of men who have a woman next to them. (b) Repeat part (a), but now assuming that the group is randomly seated at a round table.
A coin having probability p of landing on heads is flipped n times. Compute the expected number of runs of heads of size 1, of size 2, and of size k, 1 ≤ k ≤ n.
A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.
In Example 2h, say that i and j, i ≠ j, form a matched pair if i chooses the hat belonging to j and j chooses the hat belonging to i. Find the expected number of matched pairs.
A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.(a) If you are not
Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the 1st card is an ace, or the 2nd a deuce, or the 3rd a three, or . . ., or the 13th a king, or the 14 an ace, and so on, we say that a match occurs. Note that we do not require that the (13n + 1)th card be any
A certain region is inhabited by r distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type i with probability(a) Compute the mean number of insects that are caught before the first type 1 catch.(b) Compute the mean
The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three.
In an urn containing n balls, the ith ball has weight W(i), i = 1, . . . , n. The balls are removed without replacement, one at a time, according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights
For a group of 100 people, compute(a) The expected number of days of the year that are birthdays of exactly 3 people:(b) The expected number of distinct birthdays.
How many times would you expect to roll a fair die before all 6 sides appeared at least once?
A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill)
Let X1, X2, . . . be a sequence of independent and identically distributed continuous random variables. Let N ≥ 2 be such that X1 ≥ X2 ≥ . . . ≥ XN−1 < XN That is, N is the point at which the sequence stops decreasing. Show that E[N] = e. First find P{N ≥ n}.
If X1, X2, . . . ,Xn are independent and identically distributed random variables having uniform distributions over (0, 1), find(a) E[max(X1, . . . ,Xn)];(b) E[min(X1, . . . ,Xn)].
If 101 items are distributed among 10 boxes, then at least one of the boxes must contain more than 10 items. Use the probabilistic method to prove this result.
The k-of-r-out-of-n circular reliability system, k ≤ r ≤ n, consists of n components that are arranged in a circular fashion. Each component is either functional or failed, and the system functions if there is no block of r consecutive components of which at least k are failed. Show that there
There are 4 different types of coupons, the first 2 of which compose one group and the second 2 another group. Each new coupon obtained is type I with probability pi, where p1 = p2 = 1/8, p3 = p4 = 3/8. Find the expected number of coupons that one must obtain to have at least one of(a) All 4
Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find (a) P{W > 0} (b) P{W < 0} (c) E[W]
If X and Y are independent and identically distributed with mean μ and variance σ2, find E[(X − Y)2]
In Problem 6, calculate the variance of the sum of the rolls.Problem 6A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls.
In Problem 9, compute the variance of the number of empty urns.Problem 9A total of n balls, numbered 1 through n, are put into n urns, also numbered 1 through n in such a way that ball i is equally likely to go into any of the urns 1, 2, . . . , i. Find(a) The expected number of urns that are
If E[X] = 1 and Var(X) = 5, find (a) E[(2 + X)2]; (b) Var(4 + 3X). Discuss.
If 10 married couples are randomly seated at a round table, compute(a) The expected number and(b) The variance of the number of wives who are seated next to their husbands.
Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain (a) 2 aces; (b) 5 spades; (c) All 13 hearts.
Let X be the number of 1’s and Y the number of 2’s that occur in n rolls of a fair die. Compute Cov(X, Y).
A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second. Compute Cov(X, Y).
The random variables X and Y have a joint density function given byCompute Cov(X, Y).
Let X1, . . . be independent with common mean μ and common variance σ2, and set Yn = Xn + Xn+1 + Xn+2. For j ≥ 0, find Cov(Yn, Yn+j).
If X and Y have joint density functionfind(a) E[XY](b) E[X](c) E[Y]
The joint density function of X and Y is given byf(x, y) = 1/ye−(y+x/y),x > 0, y > 0Find E[X], E[Y], and show that Cov(X, Y) = 1.
A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the 20? What assumptions are you making?
A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of
Let X1, X2, . . . ,Xn be independent random variables having an unknown continuous distribution function F, and let Y1,Y2, . . . ,Ym be independent random variables having an unknown continuous distribution function G. Now order those n + m variables, and letThe random variableis the sum of the
Between two distinct methods for manufacturing certain goods, the quality of goods produced by method i is a continuous random variable having distribution Fi, i = 1, 2. Suppose that n goods are produced by method 1 and m by method 2. Rank the n + m goods according to quality, and let
If X1, X2, X3, and X4 are (pairwise) uncorrelated random variables, each having mean 0 and variance 1, compute the correlations of(a) X1 + X2 and X2 + X3;(b) X1 + X2 and X3 + X4.
Consider the following dice game, as played at a certain gambling casino: Players 1 and 2 roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Player i, i = 1, 2, wins if his roll is strictly greater than the bank€™s. For i = 1, 2,
Consider a graph having n vertices labeled 1, 2, . . . , n, and suppose that, between each of thepairs of distinct vertices, an edge is independently present with probability p. The degree of vertex i, designated as Di, is the number of edges that have vertex i as one of their vertices.(a) What is
A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find(a) E[X];(b) E[X|Y = 1];(c) E[X|Y = 5].
There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, .4 and .7. One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of

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