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

Two numbers are selected at random from the interval (0, 1). If these values are uniformly and independently distributed, by cutting the interval at these numbers, compute the probability that the three resulting line segments can form a triangle.
Let X1,X2, . . . , Xn be a random sample from a gamma distribution with known parameter α = 3 and unknown β > 0. Discuss the construction of a confidence interval for β.
When 100 tacks were thrown on a table, 60 of them landed point up. Obtain a 95% confidence interval for the probability that a tack of this type lands point up. Assume independence.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from the exponential distribution with pdf f(x) = e−x, 0 < x < ∞, zero elsewhere.(a) Show that Z1 = nY1, Z2 = (n−1)(Y2 −Y1), Z3 = (n−2)(Y3 −Y2), . . . , Zn = Yn−Yn−1 are independent and
Let two independent random variables, Y1 and Y2, with binomial distributions that have parameters n1 = n2 = 100, p1, and p2, respectively, be observed to be equal to y1 = 50 and y2 = 40. Determine an approximate 90% confidence interval for p1 − p2.
In the Program Evaluation and Review Technique (PERT), we are interested in the total time to complete a project that is comprised of a large number of subprojects. For illustration, let X1, X2, X3 be three independent random times for three subprojects. If these subprojects are in series (the
Let Y1 < Y2 < Y3 < Y4 < Y5 denote the order statistics of a random sample of size 5 from a distribution of the continuous type. Compute:(a) P(Y1 < ξ0.5 < Y5).(b) P(Y1 < ξ0.25 < Y3).(c) P(Y4 < ξ0.80 < Y5).
Compute P(Y3 < ξ0.5 < Y7) if Y1 < · · · < Y9 are the order statistics of a random sample of size 9 from a distribution of the continuous type.
Let {Xn} be a sequence of p-dimensional random vectors. Show thatfor all vectors a ∈ Rp. D XnNp(μ, Σ) if and only if a'XnN₁(a'µ, a'Σa),
Let y1 < y2 < y3 be the observed values of the order statistics of a random sample of size n = 3 from a continuous type distribution. Without knowing these values, a statistician is given these values in a random order, and she wants to select the largest; but once she refuses an observation,
Let Y1 < Y2 < · · · < Yn denote the order statistics of a random sample of size n from a distribution that has pdf f(x) = 3x2 θ3, 0 < x < θ, zero elsewhere.(a) Show that P(c < Yn/θ < 1) = 1 − c3n, where 0 < c < 1.(b) If n is 4 and if the observed value of Y4 is
Let ‾X denote the mean of a random sample of size 100 from a distribution that is χ2(50). Compute an approximate value of P(49 < ‾X < 51).
Let Xn and Yn be p-dimensional random vectors. Show that ifwhere X is a p-dimensional random vector, then Yn D→ X. P Xn - Yn 0 and XnDX,
Let {an} be a sequence of real numbers. Hence, we can also say that {an} is a sequence of constant (degenerate) random variables. Let a be a real number. Show that an → a is equivalent to an P→ a.
Let ‾X denote the mean of a random sample of size 128 from a gamma distribution with α = 2 and β = 4. Approximate P(7 < ‾X < 9).
Let X1, . . . , Xn be a random sample from a uniform(a, b) distribution. Let Y1 = min Xi and let Y2 = max Xi. Show that (Y1, Y2)' converges in probability to the vector (a, b)'.
Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) = e−(x−θ), θ < x < ∞, zero elsewhere. Let Zn = n(Y1 − θ). Investigate the limiting distribution of Zn.
Let Y denote the sum of the observations of a random sample of size 12 from a distribution having pmf p(x) = 1/6, x = 1, 2, 3, 4, 5, 6, zero elsewhere. Compute an approximate value of P(36 ≤ Y ≤ 48).
Suppose Xn has a Np(μn,Σn) distribution. Show that Χn 5 Ν,(μ, Σ) if μη → μ and Ση – Σ.
Let Xn and Yn be p-dimensional random vectors such that Xn and Yn are independent for each n and their mgfs exist. Show that ifwhere X and Y are p-dimensional random vectors, then (Xn, Yn) D→ (X,Y). XnDX and Y₂DY,
Compute an approximate probability that the mean of a random sample of size 15 from a distribution having pdf f(x) = 3x2, 0 < x < 1, zero elsewhere, is between 3/5 and 4/5.
Let the pmf of Yn be pn(y) = 1, y = n, zero elsewhere. Show that Yn does not have a limiting distribution. (In this case, the probability has “escaped” to infinity.)
Let Y be b(400, 1/5). Compute an approximate value of P(0.25 < Y/400).
Let X1, X2, . . . , Xn be a random sample of size n from a distribution that is N(μ, σ2), where σ2 > 0. Show that the sum Zn =Σn1 Xi does not have a limiting distribution.
If Y is b(100, 1/2), approximate the value of P(Y = 50).
Let f(x) = 1/x2, 1 < x < ∞, zero elsewhere, be the pdf of a random variable X. Consider a random sample of size 72 from the distribution having this pdf. Compute approximately the probability that more than 50 of the observations of the random sample are less than 3.
Forty-eight measurements are recorded to several decimal places. Each of these 48 numbers is rounded off to the nearest integer. The sum of the original 48 numbers is approximated by the sum of these integers. If we assume that the errors made by rounding off are iid and have a uniform distribution
Let the random variable Zn have a Poisson distribution with parameter μ = n. Show that the limiting distribution of the random variable Yn = (Zn−n)/√n is normal with mean zero and variance 1.
Let X1,X2, . . .,Xn be a random sample from a Poisson distribution with mean μ. Thus, Y = Σni=1 Xi has a Poisson distribution with mean nμ. Moreover, ‾X = Y/n is approximately N(μ, μ/n) for large n. Show that u(Y/n) =Y/n is a function of Y/n whose variance is essentially free of μ.
Let X1,X2, . . . , Xn be a random sample from a Γ(α = 3, β = θ) distribution, 0 < θ < ∞. Determine the mle of θ.
Let X1,X2, . . .,Xn represent a random sample from each of the distributions having the following pdfs:(a) f(x; θ) = θxθ−1, 0 < x < 1, 0 < θ < ∞, zero elsewhere.(b) f(x; θ) = e−(x−θ), θ ≤ x < ∞, −∞ < θ < ∞, zero elsewhere.
Given f(x; θ) = 1/θ, 0 < x < θ, zero elsewhere, with θ > 0, formally compute the reciprocal ofCompare this with the variance of (n+1)Yn/n, where Yn is the largest observation of a random sample of size n from this distribution. nE [alog f(X:0)] { [²¹ º]}. 20
Let X1,X2, . . . , Xn be a random sample from the distribution N(θ1, θ2). Show that the likelihood ratio principle for testing H0 : θ2 = θ'2 specified, and θ1 unspecified against H1 : θ2 ≠ θ'2, θ1 unspecified, leads to a test that rejects when Σn1 (xi − ‾x)2 ≤ c1 or Σn1 (xi −
Let X1,X2, . . . , Xn be a random sample from a N(μ0, σ2 = θ) distribution, where 0 < θ < ∞and μ0 is known. Show that the likelihood ratio test of H0 : θ = θ0 versus H1 : θ ≠ θ0 can be based upon the statistic W = Σni=1(Xi − μ0)2/θ0. Determine the null distribution of W and
Suppose X1,X2, . . . , Xn1 is a random sample from a N(θ, 1) distribution. Besides these n1 observable items, suppose there are n2 missing items, which we denote by Z1, Z2, . . ., Zn2 . Show that the first-step EM estimate iswhere ^θ(0) is an initial estimate of θ and n = n1 + n2. (1) n₁T +
For the test described in Exercise 6.3.5, obtain the distribution of the test statistic under general alternatives. If computational facilities are available, sketch this power curve for the case when θ0 = 1, n = 10, μ = 0, and α = 0.05.Exercise 6.3.5Let X1,X2, . . . , Xn be a random sample from
Consider two Bernoulli distributions with unknown parameters p1 and p2. If Y and Z equal the numbers of successes in two independent random samples, each of size n, from the respective distributions, determine the mles of p1 and p2 if we know that 0 ≤ p1 ≤ p2 ≤ 1.
The following data are observations of the random variable X = (1−W)Y1+ WY2, where W has a Bernoulli distribution with probability of success 0.70; Y1 has a N(100, 202) distribution; Y2 has a N(120, 252) distribution; W and Y1 are independent; and W and Y2 are independent.Program the EM algorithm
If X1,X2, . . . , Xn is a random sample from a distribution with pdfshow that Y = 2 ‾X is an unbiased estimator of θ and determine its efficiency. f(x; 0) = { 303 (x+0)4 0 0
Let X1,X2, . . . , Xn be a random sample from a N(θ, σ2) distribution, where σ2 is fixed but −∞ < θ < ∞.(a) Show that the mle of θ is ‾X.(b) If θ is restricted by 0 ≤ θ < ∞, show that the mle of θ is ^θ = max{0,‾X}.
Let X1,X2, . . . , Xn be a random sample from a Γ(α = 3, β = θ) distribution, where 0 < θ < ∞.(a) Show that the likelihood ratio test of H0 : θ = θ0 versus H1 : θ ≠ θ0 is based upon the statistic W = Σn i=1 Xi. Obtain the null distribution of 2W /θ0.(b) For θ0 = 3 and n =
Let X1,X2, . . .,Xn be a random sample from the Poisson distribution with 0 < θ ≤ 2. Show that the mle of θ is ^θ = min{‾X, 2}.
Let X1,X2, . . .,Xn be a random sample from a distribution with pdf f(x; θ) = θ exp{−|x|θ} /2Γ(1/θ), −∞ < x < ∞, where θ > 0. Suppose Ω = {θ : θ = 1, 2}. Consider the hypotheses H0 : θ = 2 (a normal distribution) versus H1 : θ = 1 (a double exponential distribution).
Let X1,X2, . . .,Xn be a random sample from the beta distribution with α = β = θ and Ω = {θ : θ = 1, 2}. Show that the likelihood ratio test statistic Λ for testing H0 : θ = 1 versus H1 : θ = 2 is a function of the statistic W = Σni=1 log Xi + Σni=1 log (1 − Xi).
Let S2 be the sample variance of a random sample of size n > 1 from N(μ, θ), 0 < θ < ∞, where μ is known. We know E(S2) = θ.(a) What is the efficiency of S2?(b) Under these conditions, what is the mle ^θ of θ?(c) What is the asymptotic distribution of √n(^θ − θ)?
Let X1,X2, . . .,Xn be a random sample from a Poisson distribution with mean θ > 0. Test H0 : θ = 2 against H1 : θ ≠ 2 using(a) −2 logΛ.(b) A Wald-type statistic.(c) Rao’s score statistic.
Let X1,X2, . . .,Xn be a random sample from a Γ(α, β) distribution where α is known and β > 0. Determine the likelihood ratio test for H0 : β = β0 against H1 : β ≠ β0.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from a uniform distribution on (0, θ), where θ > 0.(a) Show that Λ for testing H0 : θ = θ0 against H1 : θ ≠ θ0 is Λ = (Yn/θ0)n, Yn ≤ θ0, and Λ =0 if Yn > θ0.(b) When H0 is true, show that −2
Let the number X of accidents have a Poisson distribution with mean λθ. Suppose λ, the liability to have an accident, has, given θ, a gamma pdf with parameters α = h and β = h−1; and θ, an accident proneness factor, has a generalized Pareto pdf with parameters α, λ = h, and k. Show that
If the variance of the random variable X exists, show that E(X²) > [E(X)]².
From a well-shuffled deck of ordinary playing cards, four cards are turned over one at a time without replacement. What is the probability that the spades and red cards alternate?
Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if:(a) pX(x) = 1, x = 0, zero elsewhere.(b) pX(x) = 1/3, x = −1, 0, 1, zero elsewhere.(c) pX(x) = x/15, x = 1, 2, 3, 4, 5, zero elsewhere.
Let X be a number selected at random from a set of numbers {51, 52, . . . , 100}. Approximate E(1/X).
One of the numbers 1, 2, . . . , 6 is to be chosen by casting an unbiased die. Let this random experiment be repeated five independent times. Let the random variable X1 be the number of terminations in the set {x : x = 1, 2, 3} and let the random variable X2 be the number of terminations in the set
Show that the moment generating function of the negative binomial distribution is M(t) = pr[1 − (1 − p)et]−r. Find the mean and the variance of this distribution.
Let the mutually independent random variables X1, X2, and X3 be N(0, 1), N(2, 4), and N(−1, 1), respectively. Compute the probability that exactly two of these three variables are less than zero.
Find the uniform distribution of the continuous type on the interval (b, c) that has the same mean and the same variance as those of a chi-square distribution with 8 degrees of freedom. That is, find b and c.
Find the union C1 ∪ C2 and the intersection C1 ∩ C2 of the two sets C1 and C2, where(a) C1 = {0, 1, 2, }, C2 = {2, 3, 4}.(b) C1 = {x : 0 < x < 2}, C2 = {x : 1 ≤ x < 3}.(c) C1 = {(x, y) : 0 < x < 2, 1 < y < 2}, C2 = {(x, y) : 1 < x < 3, 1 < y < 3}.
Let a card be selected from an ordinary deck of playing cards. The outcome c is one of these 52 cards. Let X(c) = 4 if c is an ace, let X(c) = 3 if c is a king, let X(c) = 2 if c is a queen, let X(c) = 1 if c is a jack, and let X(c) = 0 otherwise. Suppose that P assigns a probability of 1/52 to
Let X be a random variable with mean μ and let E[(X − μ)2k] exist. Show, with d > 0, that P(|X − μ| ≥ d) ≤ E[(X − μ)2k]/d2k. This is essentially Chebyshev’s inequality when k = 1. The fact that this holds for all k = 1, 2, 3, . . . , when those (2k)th moments exist, usually
A positive integer from one to six is to be chosen by casting a die. Thus the elements c of the sample space C are 1, 2, 3, 4, 5, 6. Suppose C1 = {1, 2, 3, 4} and C2 = {3, 4, 5, 6}. If the probability set function P assigns a probability of 1/6 to each of the elements of C, compute P(C1), P(C2),
Let X equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of X and compute the probability that X is equal to an odd number.
Find the complement Cc of the set C with respect to the space C if(a) C = {x : 0 < x < 1}, C = {x : 5 /8 < x < 1}.(b) C = {(x, y, z) : x2 + y2 + z2 ≤ 1}, C = {(x, y, z) : x2 + y2 + z2 = 1}.(c) C = {(x, y) : |x| + |y| ≤ 2}, C = {(x, y) : x2 + y2 < 2}.
For each of the following, find the constant c so that p(x) satisfies the condition of being a pmf of one random variable X.(a) p(x) = c(2/3)x, x = 1, 2, 3, . . . , zero elsewhere.(b) p(x) = cx, x = 1, 2, 3, 4, 5, 6, zero elsewhere.
Let X be a random variable such that P(X ≤ 0) = 0 and let μ = E(X) exist. Show that P(X ≥ 2μ) ≤ 1/2.
A random experiment consists of drawing a card from an ordinary deck of 52 playing cards. Let the probability set function P assign a probability of 1/52 to each of the 52 possible outcomes. Let C1 denote the collection of the 13 hearts and let C2 denote the collection of the 4 kings. Compute
Assume that P(C1 ∩ C2 ∩ C3) > 0. Prove thatP(C1 ∩ C2 ∩ C3 ∩ C4) = P(C1)P(C2|C1)P(C3|C1 ∩ C2)P(C4|C1 ∩ C2 ∩ C3).
Suppose we are playing draw poker. We are dealt (from a well-shuffled deck) five cards, which contain four spades and another card of a different suit. We decide to discard the card of a different suit and draw one card from the remaining cards to complete a flush in spades (all five cards spades).
For each of the following distributions, compute P(μ − 2σ(a) f(x) = 6x(1 − x), 0 < x < 1, zero elsewhere.(b) p(x) = (1/2 )x, x = 1, 2, 3, . . . , zero elsewhere.
List all possible arrangements of the four letters m, a, r, and y. Let C1 be the collection of the arrangements in which y is in the last position. Let C2 be the collection of the arrangements in which m is in the first position. Find the union and the intersection of C1 and C2.
Let pX(x) = x/15, x = 1, 2, 3, 4, 5, zero elsewhere, be the pmf of X. Find P(X = 1 or 2), P(1/2 < X < 5/2), and P(1 ≤ X ≤ 2).
If X is a random variable such that E(X) = 3 and E(X2) = 13, use Chebyshev’s inequality to determine a lower bound for the probability P(−2 < X <8).
Suppose that p(x) = 1/5, x = 1, 2, 3, 4, 5, zero elsewhere, is the pmf of the discrete-type random variable X. Compute E(X) and E(X2). Use these two results to find E[(X + 2)2] by writing (X + 2)2 = X2 + 4X + 4.
A coin is to be tossed as many times as necessary to turn up one head. Thus the elements c of the sample space C are H, TH, TTH, TTTH, and so forth. Let the probability set function P assign to these elements the respective probabilities 1/2 , 1/4 , 1/8 , 1/16 , and so forth. Show that P(C) = 1.
If the sample space is C = C1 ∪ C2 and if P(C1) = 0.8 and P(C2) = 0.5, find P(C1 ∩ C2).
A hand of 13 cards is to be dealt at random and without replacement from an ordinary deck of playing cards. Find the conditional probability that there are at least three kings in the hand given that the hand contains at least two kings.
Let the random variable X have mean μ, standard deviation σ, and mgf M(t), −h < t < h. Show thatand E (*=*)=0., 5 [(**)*] =₁. (X− E 1,
Let a random variable X of the continuous type have a pdf f(x) whose graph is symmetric with respect to x = c. If the mean value of X exists, show that E(X) = c.
By the use of Venn diagrams, in which the space C is the set of points enclosed by a rectangle containing the circles C1, C2, and C3, compare the following sets. These laws are called the distributive laws.(a) C1 ∩ (C2 ∪ C3) and (C1 ∩ C2) ∪ (C1 ∩ C3).(b) C1 ∪ (C2 ∩ C3) and (C1 ∪ C2)
Let us select five cards at random and without replacement from an ordinary deck of playing cards.(a) Find the pmf of X, the number of hearts in the five cards.(b) Determine P(X ≤ 1).
Let the pmf p(x) be positive at x = −1, 0, 1 and zero elsewhere.(a) If p(0) = 1/4 , find E(X2).(b) If p(0) = 1/4 and if E(X) = 1/4 , determine p(−1) and p(1).
Let the sample space be C = {c : 0 < c < ∞}. Let C ⊂ C be defined by C = {c : 4 < c < ∞} and take P(C) = ∫C e−x dx. Show that P(C) = 1. Evaluate P(C), P(Cc), and P(C ∪ Cc).
Let X have the pdf f(x) = 3x2, 0 < x < 1, zero elsewhere. Consider a random rectangle whose sides are X and (1−X). Determine the expected value of the area of the rectangle.
Show that the moment generating function of the random variable X having the pdf f(x) = 1/3 , −1 < x < 2, zero elsewhere, is M (t) = { e2t-e- 3t 1 t #0 t = 0.
A drawer contains eight different pairs of socks. If six socks are taken at random and without replacement, compute the probability that there is at least one matching pair among these six socks.
If a sequence of sets C1, C2, C3, . . . is such that Ck ⊂ Ck+1, k = 1, 2, 3, . . . , the sequence is said to be a nondecreasing sequence. Give an example of this kind of sequence of sets.
If C1 and C2 are subsets of the sample space C, show that P(C₁ C₂) ≤ P(C₁) ≤ P(C₁ UC₂) ≤ P(C₁) + P(C₂).
Let the probability set function of the random variable X be PX(D) = ∫D f(x) dx, where f(x) = 2x/9, for x ∈ D = {x : 0 < x < 3}. Define the events D1 = {x : 0 < x < 1} and D2 = {x : 2 < x < 3}. Compute PX(D1), PX(D2), and PX(D1 ∪ D2).
Let X be a positive random variable; i.e., P(X ≤ 0) = 0. Argue that(a) E(1/X) ≥ 1/E(X)(b) E[−log X]≥ −log[E(X)](c) E[log(1/X)] ≥ log[1/E(X)](d) E[X3] ≥ [E(X)]3.
If the sample space is C = {c : −∞ < c < ∞} and if C ⊂ C is a set for which the integral ∫C e−|x| dx exists, show that this set function is not a probability set function. What constant do we multiply the integrand by to make it a probability set function?
A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation.
A pair of dice is cast until either the sum of seven or eight appears.(a) Show that the probability of a seven before an eight is 6/11.(b) Next, this pair of dice is cast until a seven appears twice or until each of a six and eight has appeared at least once. Show that the probability of the six
If a sequence of sets C1, C2, C3, . . . is such that Ck ⊃ Ck+1, k = 1, 2, 3, . . . , the sequence is said to be a nonincreasing sequence. Give an example of this kind of sequence of sets.
Let X be a random variable of the continuous type that has pdf f(x). If m is the unique median of the distribution of X and b is a real constant, show thatprovided that the expectations exist. For what value of b is E(|X −b|) a minimum? E (X — b}) = E(X − m)) + 2 [*(b − x) ƒ (x) dx, - -
Given the cdfsketch the graph of F(x) and then compute: (a) P(−1/2 < X ≤ 1/2)(b) P(X = 0)(c) P(X = 1)(d) P(2 < X ≤ 3). F(x)= = x < -1 0 2+2 +² -1
Let the space of the random variable X be D = {x : 0 < x < 1}. If D1 = {x : 0 < x < 1/2} and D2 = {x : 1/2 ≤ x < 1}, find PX(D2) if PX(D1) = 1/4.
In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce 1%, 4%, and 2% defective springs, respectively. Of the total production of springs in the factory, Machine I produces 30%, Machine II produces 25%, and Machine III produces
Let X be a random variable such that E[(X −b)2] exists for all real b. Show that E[(X − b)2] is a minimum when b = E(X).
Suppose C1, C2, C3, . . . is a nondecreasing sequence of sets, i.e., Ck ⊂ Ck+1, for k = 1, 2, 3, . . . . Then limk→∞ Ck is defined as the union C1 ∪C2 ∪C3∪· · ·. Find limk→∞ Ck if(a) Ck = {x : 1/k ≤ x ≤ 3 − 1/k}, k = 1, 2, 3, . . . .(b) Ck = {(x, y) : 1/k ≤ x2 + y2
Let C1, C2, and C3 be three mutually disjoint subsets of the sample space C. Find P[(C1 ∪ C2) ∩ C3] and P(Cc1∪ Cc2).

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