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Introduction To Mathematical Statistics 7th Edition Robert V., Joseph W. McKean, Allen T. Craig - Solutions

Bowl I contains six red chips and four blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl II, which was originally empty. One chip is then drawn at random from bowl II. Given that this chip is blue, find the conditional probability that two red chips
If C1, C2, C3, . . . are sets such that Ck ⊃ Ck+1, k = 1, 2, 3, . . ., limk→∞ Ck is defined as the intersection C1 ∩ C2 ∩ C3 ∩ · · · . Find limk→∞ Ck if(a) Ck = {x : 2 − 1/k < x ≤ 2}, k = 1, 2, 3, . . . .(b) Ck = {x : 2 < x ≤ 2 + 1/k}, k = 1, 2, 3, . . . .(c) Ck =
Consider an urn which contains slips of paper each with one of the numbers 1, 2, . . . , 100 on it. Suppose there are i slips with the number i on it for i = 1, 2, . . . , 100. For example, there are 25 slips of paper with the number 25. Assume that the slips are identical except for the numbers.
In an office there are two boxes of computer disks: Box C1 contains seven Verbatim disks and three Control Data disks, and box C2 contains two Verbatim disks and eight Control Data disks. A person is handed a box at random with prior probabilities P(C1) = 2/3 and P(C2) = 1/3, possibly due to the
For every one-dimensional set C, define the function Q(C) = ΣC f(x), where f(x) = (2/3)(1/3)x, x = 0, 1, 2, . . . , zero elsewhere. If C1 = {x : x = 0, 1, 2, 3} and C2 = {x : x = 0, 1, 2, . . .}, find Q(C1) and Q(C2).
Let X denote a random variable for which E[(X − a)2] exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.
A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If four chips are taken at random and without replacement, find the probability that:(a) Each of the four chips is red; (b) None of the four chips is red; (c) There is at least one chip of each color.
A person has purchased 10 of 1000 tickets sold in a certain raffle. To determine the five prize winners, five tickets are to be drawn at random and without replacement. Compute the probability that this person wins at least one prize.
If C1 and C2 are independent events, show that the following pairs of events are also independent: (a) C1 and Cc2(b) Cc1 and C2(c) Cc1 and Cc2.
For every one-dimensional set C for which the integral exists, let Q(C) = ∫C f(x) dx, where f(x) = 6x(1 − x), 0 < x < 1, zero elsewhere; otherwise, let Q(C) be undefined. If C1 = {x : 1/4 < x < 3/4 }, C2 = {1/2}, and C3 = {x : 0 < x < 10}, find Q(C1),Q(C2), and Q(C3).
Let X denote a random variable such that K(t) = E(tX) exists for all real values of t in a certain open interval that includes the point t = 1. Show that K(m)(1) is equal to the mth factorial moment E[X(X − 1) · · · (X − m + 1)].
For each of the following cdfs F(x), find the pdf f(x) [pmf in part (d)], the 25th percentile, and the 60th percentile. Also, sketch the graphs of f(x) and F(x).(a) F(x) = (1+e−x)−1 ,−∞ < x < ∞.(b) F(x) = exp{−e−x} ,−∞ < x < ∞.(c) F(x) = 1/2 + 1/π tan−1(x) ,−∞
Compute the probability of being dealt at random and without replacement a 13-card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club(b) 13 cards of the same suit.
Let C1 and C2 be independent events with P(C1) = 0.6 and P(C2) = 0.3. Compute (a) P(C1 ∩ C2) (b) P(C1 ∪ C2) (c) P(C1 ∪ Cc2).
For every two-dimensional set C contained in R2 for which the integral exists, let Q(C) = ∫ ∫C(x2 + y2) dxdy. If C1 = {(x, y) : −1 ≤ x ≤ 1,−1 ≤ y ≤ 1}, C2 = {(x, y) : −1 ≤ x = y ≤ 1}, and C3 = {(x, y) : x2+y2 ≤ 1}, find Q(C1),Q(C2), and Q(C3).
Let X be a random variable. If m is a positive integer, the expectation E[(X − b)m], if it exists, is called the mth moment of the distribution about the point b. Let the first, second, and third moments of the distribution about the point 7 be 3, 11, and 15, respectively. Determine the mean μ
Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) Their sum is even(b) Their product is even.
Let C denote the set of points that are interior to, or on the boundary of, a square with opposite vertices at the points (0, 0) and (1, 1). Let Q(C) = ∫ ∫C dy dx.(a) If C ⊂ C is the set {(x, y) : 0 < x < y < 1}, compute Q(C).(b) If C ⊂ C is the set {(x, y) : 0 < x = y < 1},
Let X be a random variable such that R(t) = E(et(X−b)) exists for t such that −h < t < h. If m is a positive integer, show that R(m)(0) is equal to the mth moment of the distribution about the point b.
There are five red chips and three blue chips in a bowl. The red chips are numbered 1, 2, 3, 4, 5, respectively, and the blue chips are numbered 1, 2, 3, respectively. If two chips are to be drawn at random and without replacement, find the probability that these chips have either the same number
Each of four persons fires one shot at a target. Let Ck denote the event that the target is hit by person k, k = 1, 2, 3, 4. If C1, C2, C3, C4 are independent and if P(C1) = P(C2) = 0.7, P(C3) = 0.9, and P(C4) = 0.4, compute the probability that (a) All of them hit the target(b) Exactly one
Let C be the set of points interior to or on the boundary of a cube with edge of length 1. Moreover, say that the cube is in the first octant with one vertex at the point (0, 0, 0) and an opposite vertex at the point (1, 1, 1). Let Q(C) ∫ ∫ ∫C dxdydz.(a) If C ⊂ C is the set {(x, y, z) : 0
Let X be a random variable with mean μ and variance σ2 such that the third moment E[(X − μ)3] about the vertical line through μ exists. The value of the ratio E[(X − μ)3]/σ3 is often used as a measure of skewness. Graph each of the following probability density functions and show that
Let C denote the set {(x, y, z) : x2 + y2 + z2 ≤ 1}. Using spherical coordinates, evaluate Q(C)= +zphpap z²+z²+z²^²√ √ √ = (
Let X have the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Compute the probability that X is at least 3/4 given that X is at least 1/2.
In a lot of 50 light bulbs, there are 2 bad bulbs. An inspector examines five bulbs, which are selected at random and without replacement.(a) Find the probability of at least one defective bulb among the five.(b) How many bulbs should be examined so that the probability of finding at least one bad
A bowl contains three red (R) balls and seven white (W) balls of exactly the same size and shape. Select balls successively at random and with replacement so that the events of white on the first trial, white on the second, and so on, can be assumed to be independent. In four trials, make certain
Let X be a random variable with mean μ and variance σ2 such that the fourth moment E[(X − μ)4] exists. The value of the ratio E[(X − μ)4]/σ4 is often used as a measure of kurtosis. Graph each of the following probability density functions and show that this measure is smaller for the first
A coin is tossed two independent times, each resulting in a tail (T) or a head (H). The sample space consists of four ordered pairs: TT, TH, HT, HH. Making certain assumptions, compute the probability of each of these ordered pairs. What is the probability of at least one head?
To join a certain club, a person must be either a statistician or a mathematician or both. Of the 25 members in this club, 19 are statisticians and 16 are mathematicians. How many persons in the club are both a statistician and a mathematician?
A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct envelope?
After a hard-fought football game, it was reported that, of the 11 starting players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm, 2 hurt both a hip and a knee, 1 hurt both an arm and a knee, and no one hurt all three. Comment on the accuracy of the report.
A die is cast independently until the first 6 appears. If the casting stops on an odd number of times, Bob wins; otherwise, Joe wins.(a) Assuming the die is fair, what is the probability that Bob wins?(b) Let p denote the probability of a 6. Show that the game favors Bob, for all p, 0 < p < 1.
Let X be the number of gallons of ice cream that is requested at a certain store on a hot summer day. Assume that f(x) = 12x(1000−x)2/1012, 0 < x < 1000, zero elsewhere, is the pdf of X. How many gallons of ice cream should the store have on hand each of these days, so that the probability
Cards are drawn at random and with replacement from an ordinary deck of 52 cards until a spade appears.(a) What is the probability that at least four draws are necessary?(b) Same as part (a), except the cards are drawn without replacement.
Suppose the experiment is to choose a real number at random in the interval (0, 1). For any subinterval (a, b) ⊂ (0, 1), it seems reasonable to assign the probability P[(a, b)] = b−a; i.e., the probability of selecting the point from a subinterval is directly proportional to the length of the
A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at least one is correct?
Suppose a fair 6-sided die is rolled six independent times. A match occurs if side i is observed on the ith trial, i = 1, . . . , 6.(a) What is the probability of at least one match on the six rolls?(b) Extend part (a) to a fair n-sided die with n independent rolls. Then determine the limit of the
Suppose D is a nonempty collection of subsets of C. Consider the collection of events B=n{E: DCE and & is a o-field}.
Consider the events C1, C2, C3.(a) Suppose C1, C2, C3 are mutually exclusive events. If P(Ci) = pi, i = 1, 2, 3, what is the restriction on the sum p1 + p2 + p3?(b) In the notation of part (a), if p1 = 4/10, p2 = 3/10, and p3 = 5/10, are C1, C2, C3 mutually exclusive?For the last two exercises it
Let C = R, where R is the set of all real numbers. Let I be the set of all open intervals in R. The Borel σ-field on the real line is given byBy definition, B0 contains the open intervals. Because [a,∞) = (−∞, a)c and B0 is closed under complements, it contains all intervals of the form
If the pdf of X is f(x) = 2xe−x2/9, 0 < x < ∞, zero elsewhere, determine the pdf of Y = X2.
Let X have the cdf F(x) that is a mixture of the continuous and discrete types, namelyDetermine reasonable definitions of μ = E(X) and σ2 = var(X) and compute each. F(x)= 0 2+1 x < 0 #+¹ 0
Players A and B play a sequence of independent games. Player A throws a die first and wins on a “six.” If he fails, B throws and wins on a “five” or “six.” If he fails, A throws and wins on a “four,” “five,” or “six.” And so on. Find the probability of each player winning
Let C1, C2, C3 be independent events with probabilities 1/2 , 1/3 , 1/4 , respectively. Compute P(C1 ∪ C2 ∪ C3).
From a bowl containing five red, three white, and seven blue chips, select four at random and without replacement. Compute the conditional probability of one red, zero white, and three blue chips, given that there are at least three blue chips in this sample of four chips.
Consider k continuous-type distributions with the following characteristics: pdf fi(x), mean μi, and variance σ2i , i = 1, 2, . . . , k. If ci ≥ 0, i = 1, 2, . . . , k, and c1+c2+· · ·+ck = 1, show that the mean and the variance of the distribution having pdf c1f1(x) + · · · + ckfk(x) are
Let the three mutually independent events C1, C2, and C3 be such that P(C1) = P(C2) = P(C3) = 1/4. Find P[(Cc1 ∩ Cc2) ∪ C3].
Person A tosses a coin and then person B rolls a die. This is repeated independently until a head or one of the numbers 1, 2, 3,4 appears, at which time the game is stopped. Person A wins with the head and B wins with one of the numbers 1, 2, 3, 4. Compute the probability that A wins the game.
Hunters A and B shoot at a target; the probabilities of hitting the target are p1 and p2, respectively. Assuming independence, can p1 and p2 be selected so that P(zero hits) = P(one hit) = P(two hits)?
Each bag in a large box contains 25 tulip bulbs. It is known that 60% of the bags contain bulbs for 5 red and 20 yellow tulips, while the remaining 40% of the bags contain bulbs for 15 red and 10 yellow tulips. A bag is selected at random and a bulb taken at random from this bag is planted.(a) What
A bowl contains 10 chips numbered 1, 2, . . . , 10, respectively. Five chips are drawn at random, one at a time, and without replacement. What is the probability that two even-numbered chips are drawn and they occur on even numbered draws?
Let X1,X2,X3 be iid, each with the distribution having pdf f(x) = e−x, 0 < x < ∞, zero elsewhere. Show thatare mutually independent. Y₁ X₁ X₁ + X₂¹ Y₂ = X₁ + X₂ X₁ + X2 + X3' Y3 = X₁ + X2 + X3
A person bets 1 dollar to b dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. Find b so that the bet is fair.
Suppose there are three curtains. Behind one curtain there is a nice prize, while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then Monte Hall opens one of the other two curtains to reveal a worthless prize. Hall then expresses the willingness to
Show that the random variables X1 and X2 with joint pdfare independent. › = { ő 12x12 (12) 0
A French nobleman, Chevalier de M´er´e, had asked a famous mathematician, Pascal, to explain why the following two probabilities were different (the difference had been noted from playing the game many times): (1) At least one six in four independent casts of a six-sided die; (2) At
A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability 0.90; however, this test indicates that an impurity is there when it is not about 5% of the time. The chemist produces compounds with the impurity about 20% of
Let the random variables X and Y have the joint pmf(a) p(x, y) = 1/3, (x, y) = (0, 0), (1, 1), (2, 2), zero elsewhere.(b) p(x, y) = 1/3, (x, y) = (0, 2), (1, 1), (2, 0), zero elsewhere.(c) p(x, y) = 1/3, (x, y) = (0, 0), (1, 1), (2, 0), zero elsewhere.In each case compute the correlation
Let f(x1, x2, x3) = exp[−(x1 + x2 + x3)], 0 < x1 < ∞, 0 < x2 < ∞, 0 < x3 < ∞, zero elsewhere, be the joint pdf of X1, X2, X3.(a) Compute P(X1 < X2 < X3) and P(X1 = X2 < X3).(b) Determine the joint mgf of X1,X2, and X3. Are these random variables independent?
Let f1|2(x1|x2) = c1x1/x22 , 0 < x1 < x2, 0 < x2 < 1, zero elsewhere, and f2(x2) = c2x42 , 0 < x2 < 1, zero elsewhere, denote, respectively, the conditional pdf of X1, given X2 = x2, and the marginal pdf of X2. Determine:(a) The constants c1 and c2.(b) The joint pdf of X1 and
Let F(x, y) be the distribution function of X and Y . For all real constants a < b, c < d, show that P(a < X ≤ b, c < Y ≤ d) = F(b, d) − F(b, c) − F(a, d) + F(a, c).
Let X1 and X2 be two independent random variables so that the variances of X1 and X2 are σ21 = k and σ22 = 2, respectively. Given that the variance of Y = 3X2 − X1 is 25, find k.
Let f(x, y) = 2, 0 < x < y, 0 < y < 1, zero elsewhere, be the joint pdf of X and Y . Show that the conditional means are, respectively, (1+x)/2, 0 < x < 1, and y/2, 0 < y < 1. Show that the correlation coefficient of X and Y is ρ = 1/2.
Show that the function F(x, y) that is equal to 1 provided that x + 2y ≥ 1, and that is equal to zero provided that x+2y < 1, cannot be a distribution function of two random variables.
Let the joint pdf of X and Y be given by(a) Compute the marginal pdf of X and the conditional pdf of Y , given X = x.(b) For a fixed X = x, compute E(1 + x + Y |x) and use the result to compute E(Y |x). 2 (1+r+y f(x, y) = ={₁ ;, y) = 0
If the independent variables X1 and X2 have means μ1, μ2 and variances σ21, σ22, respectively, show that the mean and variance of the product Y = X1X2 are μ1μ2 and σ21σ22 + μ21σ22 + μ22σ21, respectively.
Suppose X1 and X2 are random variables of the discrete type which have the joint pmf p(x1, x2) = (x1 + 2x2)/18, (x1, x2) = (1,1), (1,2), (2,1), (2,2), zero elsewhere. Determine the conditional mean and variance of X2, given X1 = x1, for x1 = 1 or 2. Also, compute E(3X1 − 2X2).
If f(x1, x2) = e−x1−x2 , 0 < x1 < ∞, 0 < x2 < ∞, zero elsewhere, is the joint pdf of the random variables X1 and X2, show that X1 and X2 are independent and that M(t1, t2) = (1 − t1)−1(1 − t2)−1, t2 < 1, t1 < 1. Also show thatAccordingly, find the mean and the
Let f(x, y) = e−x−y, 0 < x < ∞, 0 < y < ∞, zero elsewhere, be the pdf of X and Y. Then if Z = X +Y , compute P(Z ≤ 0), P(Z ≤ 6), and, more generally, P(Z ≤ z), for 0 < z < ∞. What is the pdf of Z?
Let X1,X2, and X3 be three random variables with means, variances, and correlation coefficients, denoted by μ1, μ2, μ3; σ21, σ22, σ23; and ρ12, ρ13, ρ23, respectively. For constants b2 and b3, suppose E(X1−μ1|x2, x3) = b2(x2−μ2)+b3(x3−μ3). Determine b2 and b3 in terms of the
Determine the mean and variance of the sample mean ¯X = 5−1Σ5i=1 Xi, where X1, . . . , X5 is a random sample from a distribution having pdf f(x) = 4x3, 0 < x < 1, zero elsewhere.
Let X and Y have the joint pdf f(x, y) = 1, −x < y < x, 0 < x < 1, zero elsewhere. Show that, on the set of positive probability density, the graph of E(Y |x) is a straight line, whereas that of E(X|y) is not a straight line.
Let 13 cards be taken, at random and without replacement, from an ordinary deck of playing cards. If X is the number of spades in these 13 cards, find the pmf of X. If, in addition, Y is the number of hearts in these 13 cards, find the probability P(X = 2, Y = 5). What is the joint pmf of X and Y?
Let X and Y be independent random variables with means μ1, μ2 and variances σ21, σ22. Determine the correlation coefficient of X and Z = X − Y in terms of μ1, μ2, σ21, σ22.
Let X and Y have the joint pdf f(x, y) = 3x, 0 < y < x < 1, zero elsewhere. Are X and Y independent? If not, find E(X|y).
Let X1,X2,X3 be iid with common pdf f(x) = exp(−x), 0 < x < ∞, zero elsewhere. Evaluate:(a) P(X1 < X2|X1 < 2X2).(b) P(X1 < X2 < X3|X3 < 1).
Let μ and σ2 denote the mean and variance of the random variable X. Let Y = c+ bX, where b and c are real constants. Show that the mean and variance of Y are, respectively, c + bμ and b2σ2.
Let σ21 = σ22 = σ2 be the common variance of X1 and X2 and let ρ be the correlation coefficient of X1 and X2. Show for k > 0 that P(X1-1) + (X2-μ₂)| ≥ko] ≤ 2(1 + p) k2
Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y) = 2, and var(X + 2Y) = 15.
Let X and Y be random variables with the space consisting of the four points (0, 0), (1, 1), (1, 0), (1,−1). Assign positive probabilities to these four points so that the correlation coefficient is equal to zero. Are X and Y independent?
Let X and Y be random variables with means μ1, μ2; variances σ21, σ22; and correlation coefficient ρ. Show that the correlation coefficient of W = aX+b, a > 0, and Z = cY + d, c > 0, is ρ.
Two line segments, each of length two units, are placed along the x-axis. The midpoint of the first is between x = 0 and x = 14 and that of the second is between x = 6 and x = 20. Assuming independence and uniform distributions for these midpoints, find the probability that the line segments
A person rolls a die, tosses a coin, and draws a card from an ordinary deck. He receives $3 for each point up on the die, $10 for a head and $0 for a tail, and $1 for each spot on the card (jack = 11, queen = 12, king = 13). If we assume that the three random variables involved are independent and
Cast a fair die and let X = 0 if 1, 2, or 3 spots appear, let X = 1 if 4 or 5 spots appear, and let X = 2 if 6 spots appear. Do this two independent times, obtaining X1 and X2. Calculate P(|X1 − X2| = 1).
Let X1,X2 be two random variables with joint pdf f(x1, x2) = x1 exp{−x2}, for 0 < x1 < x2 < ∞, zero elsewhere. Determine the joint mgf of X1,X2. Does M(t1, t2) = M(t1, 0)M(0, t2)?
If M(t1, t2) is the mgf of a bivariate normal distribution, compute the covariance by using the formulaNow let ψ(t1, t2) = log M(t1, t2). Show that ∂2ψ(0, 0)/∂t1∂t2 gives this covariance directly. ²M (0,0) Ət1 Ət₂ M (0,0) Əti M (0,0) Ət₂
The mgf of a random variable X is ( 2/3 + 1/3 et)9. Show that r=1 ( 1 ) ¸ (²) (3) 3 = ( 08 + ¹1 > x > °7 - id I 9-x
If X is b(n, p), show that E (A) = P and E[(4->)]; Ε P p(1-P) n
Let X and Y have the parameters μ1, μ2, σ21, σ22, and ρ. Show that the correlation coefficient of X and [Y − ρ(σ2/σ1)X] is zero.
Let S2 be the sample variance of a random sample from a distribution with variance σ2 > 0. Since E(S2) = σ2, why isn’t E(S) = σ?
If X is χ2(5), determine the constants c and d so that P(c < X < d) = 0.95 and P(X < c) = 0.025.
The mgf of a random variable X is e4(et−1). Show that P(μ − 2σ < X < μ + 2σ) = 0.931.
Let X have the conditional geometric pmf θ(1−θ)x−1, x = 1, 2, . . ., where θ is a value of a random variable having a beta pdf with parameters α and β. Show that the marginal (unconditional) pmf of X isIf α = 1, we obtainwhich is one form of Zipf ’s law. T(a + 3)(a + 1)(3+x-1) r(a)(3)(a
In a lengthy manuscript, it is discovered that only 13.5 percent of the pages contain no typing errors. If we assume that the number of errors per page is a random variable with a Poisson distribution, find the percentage of pages that have exactly one error.
Let U and V be independent random variables, each having a standard normal distribution. Show that the mgf E(et(UV)) of the random variable UV is (1 − t2)−1/2, −1 < t < 1.
Assuming a computer is available, plot the pdfs of the random variables defined in parts (a)–(e) below. Obtain an overlay plot of all five pdfs, also. In R the domain values of the pdfs can easily be obtained by using the seq command. For instance, the command x<-seq(-6,6,.1) returns in x a
Suppose that g(x, 0) = 0 and thatfor x = 1, 2, 3, . . .. If g(0, w) = e−λw, show by mathematical induction that Du g(x, w)]=-Ag(x, w) + Ag(x-1, w)
Let X be a random variable such that E(Xm) = (m+1)!2m, m = 1, 2, 3, . . . . Determine the mgf and the distribution of X.
Let X have a Poisson distribution with μ = 100. Use Chebyshev’s inequality to determine a lower bound for P(75 < X < 125).

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