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theory of probability

Questions and Answers of Theory Of Probability

=+27. Let X be a random variable with moment generating function M(t)=E etX = ∞n=0μn n!t n100 4. Combinatorics defined in some neighborhood of the origin. Here μn = E(Xn) is the nth moment of X.
=+26. Consider the n-dimensional unit cube [0, 1]n. Suppose that each of its n2n−1 edges is independently assigned one of two equally likely orientations. Let S be the number of vertices at which
=+25. Consider a random graph with n nodes. Between every pair of nodes, we independently introduce an edge with probability p. A trio of nodes forms a triangle if each of its three pairs is
=+24. Balls are randomly extracted one by one from a box containing b black balls and w white balls. Show that the expected number of black balls left when the last white ball is extracted equals b
=+23. Suppose π is a random permutation of {1,...,n}. Show that E* n−1 j=1[π(j) − π(j + 1)]2+=n + 1 3.(Hint: Each term has the same expectation [139].)
=+22. Let X be the number of fixed points of a random permutation of{1,...,n}. Demonstrate that E(Xj ) =min{j,n}k=1*j k+.(Hint: Find the binomial moments of X via equation (4.34), convert these to
=+where |S| is the number of elements of the subset S of {1,...,n}.Equating coefficients of (u − 1)j in these two representations of G(u)yields the identity EX j = |S|=j Pr i∈S Ai.
=+21. Suppose the random variable X is nonnegative, bounded, and integer valued. Show that the probability generating function G(u) = E(uX)of X can be expressed as G(u) = ∞j=0 EX j (u − 1)j
=+20. In our discussion of Stirling numbers of the first kind, we showed that the number of left-to-right maxima Yn of a random permutation has the decomposition Yn = Z1 +···+Zn, where the Zk are
=+19. Prove the inequality n k≥ (n k) for all n and k by invoking the definitions of the two kinds of Stirling numbers.
=+18. Demonstrate that the harmonic number Hn = 1+ 1 2 +···+ 1 n satisfies Hn = 1 n!n+1 2. (Hint: Apply equation (4.23) and Hn = Hn−1 + 1 n .)
=+17. Show that n n= (n n) = 1, n n−1= ( n n−1) = n 2, and (n 2) = 2n−1 −1.
=+16. List in canonical form the 11 permutations of {1, 2, 3, 4} with 2 cycles.
=+15. Consider a simple random walk of 2n steps. Conditional on the event that the walk returns to 0 at step 2n, show that this is the first return with probability 1/(2n−1). (Hint: The first and
=+14. Show that the Catalan numbers satisfy the recurrence cn+1 = 2(2n + 1)n + 2 cn, which is consistent with expression (4.17).
=+13. Suppose the n random variables X1,...,Xn are independent and share the common distribution function F(x). Prove that the jth order statistic X(j) has distribution function F(j)(x) = n k=jn
=+12. Suppose that each of the random variables X1,...,Xn of Proposition 4.4.1 satisfies E(|Xi|k) < ∞. Show that all of the expectations E(|X(i)|k), E(|XS|k), and E(|XS|k) are finite. (Hints:
=+These recurrences provide an alternative to the method of inclusionexclusion [157]. Show how they can be applied to find the distribution function of the order statistic X(m) from a sample
=+4.10 Problems 97 11. Let C1,...,Cn be independent events. Define Am n to be the event that at least m of these events occur and Bm n be the event that exactly m of these events occur. Demonstrate
=+What is the probability measure and what are the events? (Hint:Consider n labeled balls falling into k labeled boxes.)
=+10. Give an inclusion-exclusion proof of the identity*n k+= 1 k!k j=0(−1)jk j(k − j)n.
=+9. Calculate the probability p[k] that exactly k suits are missing in a poker hand [59]. To a good approximation p[0] = .264, p[1] = .588, p[2] = .146, and p[3] = .002. (Hint: Each hand has
=+8. Define qr to be the probability that in r tosses of two dice each pair(1, 1),...,(6, 6) appears at least once [59]. Show that qr = 6 k=06 k(−1)k36 − k 36 r.
=+7. A permutation that satisfies the equation π(π(i)) = i for all i is called an involution [139]. Prove that a random permutation of {1,...,n}is an involution with probability n 2k=0 12kk!(n
=+6. Prove that there are 8!8 k=0(−1)k k! ways of placing eight rooks on a chessboard so that none can take another and none stands on a white diagonal square [59]. (Hint: Think of the rook
=+5. You have 10 pairs of shoes jumbled together in your closet [59]. Show that if you reach in and randomly pull out four shoes, then the probability of extracting at least one pair is 99/323.
=+96 4. Combinatorics for a and b positive reals and l and n positive integers. Note that the special case a = b = 1 amounts tol−1 k=0l kn j=1 jk = (n + 1)l − 1.(Hints: Drop l random points
=+E(Yn−1,k)and initial conditions E(Xkk) = 1 and E(Ykk) = k. Prove that these recurrences have the unique solutions E(Xnk) = n + 1 k + 1 , E(Ynk) = k(n + 1)k + 1 .4. Prove Pascal’s identityn
=+3. Suppose you select k balls randomly from a box containing n balls labeled 1 through n. Let Xnk be the lowest label selected and Ynk be the highest label selected. Demonstrate the mean
=+2. Prove the identity n−1 m=1 m m! = n! − 1 by a counting argument. (Hint: Let n − m be the first integer not fixed by a permutation π. Thus, π(i) = i for 1 ≤ i ≤ n − m − 1.)
=+Avoid algebra as much as possible, but impose reasonable restrictions on the integers k, m, and n. (Hint: Think of forming a committee of a given size from a class of a given size. You may have to
=+1. Prove the following binomial coefficient identities by constructing an appropriate bijection:n k= n n − kkn k= nn − 1 k − 14.10 Problems 95n k k m=n mn − m k − mm + n
=+24. Suppose X is a random variable satisfying 0 < a ≤ X ≤ b < ∞. Use l’Hˆopital’s rule to prove that the weighted mean M(p) = E(Xp)1 p is continuous at p = 0 if we define M(0) = eE(ln X)
=+p . Now prove Minkowski’s triangle inequality ||X+Y ||p ≤ ||X||p+||Y ||p. (Hint: Apply H¨older’s inequality to the right-hand side of E(|X + Y |p) ≤ E(|X|·|X + Y |p−1) + E(|Y |·|X + Y
=+23. Suppose 1 ≤ p < ∞. For a random variable X with E(|X|p) < ∞, define the norm ||X||p = E(Xp)1
=+22. Let f(x) be a convex function on [0, 1]. Prove that the Bernstein polynomial of degree n approximating f(x) is also convex. (Hint:Show that d2 dx2 EfSn n = n(n − 1)%EfSn−2 + 2 n−2 E
=+21. Suppose the function f(x) has continuous derivative f(x). For δ > 0 show that Bernstein’s polynomial satisfies the bound EfSn n − f(x)≤ δ||f||∞ + ||f||∞2nδ2 .Conclude from this
=+20. Let Bnf(x) = E[f(Sn/n)] denote the Bernstein polynomial of degree n approximating f(x) as discussed in Example 3.5.1. Prove that(a) Bnf(x) is linear in f(x),(b) Bnf(x) ≥ 0 if f(x) ≥ 0,(c)
=+19. Let X be a Poisson random variable with mean λ. Demonstrate that the Chernoff bound Pr(X ≥c) ≤ inf t>0 e−ct E(etX)amounts to Pr(X ≥c) ≤ (λe)c cc e−λfor any integer c>λ.
=+18. Suppose g(x) is a function such that g(x) ≤ 1 for all x and g(x) ≤ 0 for x ≤c. Demonstrate the inequality Pr(X ≥c) ≥ E[g(X)] (3.10)72 3. Convexity, Optimization, and Inequalities for
=+17. Let X be a random variable with E(X) = 0 and E(X2) = σ2. Show that Pr(X ≥c) ≤a2 + σ2(a + c)2 (3.9)for all nonnegative a andc. Prove that the choice a = σ2/c minimizes the right-hand side
=+16. If the random variable X has values in the interval [a, b], then show that Var(X) ≤ (b − a)2/4 and that this bound is sharp. (Hints: Reduce to the case [a, b] = [0, 1]. If E(X) = p, then
=+15. Suppose the random variables X and Y have densities f(u) and g(u)such that f(u) ≥ g(u) for u ≤ a and f(u) ≤ g(u) for u>a. Prove that E(X) ≤ E(Y ). If in addition f(u) = g(u) = 0 for u <
=+[148]. What do you deduce in the special cases where the family is binomial, Poisson, and exponential?
=+14. Let X1,...,Xn be n independent random variables from a common distributional family. Suppose the variance σ2(μ) of a generic member of this family is a function of the mean μ. Now consider
=+13. Suppose that in Example 3.4.2 we minimize the function h (θ) = p i=1⎧⎨⎩yi −q j=1 xij θj2+⎫⎬⎭1/2 instead of h(θ) for a small, positive number . Show that the same MM
=+12. Show that the loglikelihood L(r) in Example 3.4.1 is concave under the reparameterization ri = eθi .3.6 Problems 71
=+11. Let Hn = 1+ 1 2 +···+ 1 n . Using inequality (3.8), verify the inequality n√n n + 1 ≤ n + Hn for any positive integer n (Putnam Competition, 1975).
=+10. Heron’s classical formula for the area of a triangle with sides of lengtha,b, and c is s(s − a)(s − b)(s − c), where s = (a + b + c)/2 is the semiperimeter. Using inequality (3.8),
=+9. If f(x) is a nondecreasing function on the interval [a, b], then show that g(x) = x a f(y) dy is a convex function on [a, b].
=+8. Let f(x) be a continuous function on the real line satisfying f1 2(x + y)≤1 2f(x) + 1 2f(y).Prove that f(x) is convex.
=+7. Suppose that f(x) is a convex function on the real line. If a and y are vectors in Rm, then show that f(at y) is a convex function of y.For m > 1 show that f(at y) is not strictly convex.
=+6. Prove the unproved assertions of Proposition 3.2.4.
=+5. Prove parts (b), (c), and (d) of Proposition 3.2.3.
=+4. Prove the strict convexity assertions of Proposition 3.2.1.
=+3. Demonstrate that the function f(x) = xn−na ln x is convex on (0, ∞)for a > 0. Where does its minimum occur?
=+2. Show that Riemann’s zeta functionζ(s) = ∞n=1 1ns is log-convex for s > 1.
=+1. On which intervals are the following functions convex: ex, e−x, xn,|x|p for p ≥ 1, √1 + x2, x ln x, and cosh x? On these intervals, which functions are log-convex?
=+are independent. What distributions to these random variables follow? (Hints: Denote the random variables Z1,...,Zn. Make a multidimensional change of variables to find their joint density using
=+33. Continuing Problem 32, calculate E(Yi), Var(Yi), and Cov(Yi, Yj ) for i = j. Also show that (Y1 + Y2, Y3,...,Yn)t has a Dirichlet distribution.2.10 Problems 53 34. Continuing Problem 32,
=+32. One can generate the Dirichlet distribution by a different mechanism than the one developed in the text [114]. Take n independent gamma random variables X1,...,Xn of unit scale and form the
=+See the article [158] for explicit evaluation of the last one-dimensional integral. (Hints: Show that E[(vt X)−m] = Γ(n)Γ(n − m)Rn+(vt x)−mxm 11xm 1e−x1 dx.Then let sn−1 = n i=2
=+31. Suppose the random vector X is uniformly distributed on the unit simplex Tn−1. Let m be a positive integer with m
=+30. An Epanechnikov random vector X has density f(x) = n + 2 2vn1 − x2supported on the unit ball {x ∈ Rn : x ≤ 1}. Here vn is the volume of the ball as given in Problem 29. Demonstrate
=+29. Demonstrate that the unit ball {x ∈ Rn : x ≤ 1} has volumeπn/2/Γ(n/2 + 1) and the standard simplex {x ∈ Rn+ : x1 ≤ 1} has volume 1/n!.52 2. Calculation of Expectations
=+28. Consider a negative binomial random variable X with density Pr(X = k) = k − 1 n − 1pnqk−n for q = 1 − p and k ≥ n. Prove that for any function f(x)E[qf(X)] = E(X − n)f(X − 1)X
=+27. Suppose X has a binomial distribution with success probability p over n trials. Show that E[Xf(X)] = p 1 − p E(n − X)f(X + 1)for any function f(x). Use this identity to calculate the mean
=+26. Let χ2 n and χ2 n+2 be chi-square random variables with n and n + 2 degrees of freedom, respectively. Demonstrate that E[f(χ2 n)] = n Ef(χ2 n+2)χ2 n+2for any well-behaved function f(x)
=+25. Suppose X has a binomial distribution with success probability p over n trials. Show that E 1 X + 1= 1 − (1 − p)n+1(n + 1)p .
=+24. Let X be a nonnegative integer-valued random variable with probability generating function G(u). Prove that E 1(X + k + j)(X + k + j − 1)···(X + k)= 1 j! 1 0uk−1(1 − u)jG(u) du by
=+23. Let the positive random variable X have Laplace transform L(t).Prove that E[(aX + b)−1] = ∞0 e−btL(at) dt for a ≥ 0 and b > 0.2.10 Problems 51
=+22. Suppose the right-tail probability of a nonnegative random variable X satisfies |1 − F(x)| ≤ cx−n− for all sufficiently large x, where n is a positive integer, and and c are positive
=+21. Card matching is one way of testing extrasensory perception (ESP).The tester shuffles a deck of cards labeled 1 through n and turns cards up one by one. The subject is asked to guess the value
=+20. Calculate the Laplace transform of the probability density 1 + a2 a2 e−x[1 − cos(ax)]1{x≥0}.
=+50 2. Calculation of Expectations moments of all orders, and it is possible to represent the characteristic function of X by the series EeitX = ∞n=0 E(Xn)(it)n n! .If one can demonstrate that
=+19. Example 2.4.6 shows that it is impossible to write a random variable U uniformly distributed on [−1, 1] as the difference of two i.i.d. random variables X and Y . It is also true that it is
=+18. Show that the bilateral exponential density 1 2 e−|x| has characteristic function 1/(1 + t 2). Use this fact to calculate its mean and variance.
=+17. Let X have the gamma density defined in Problem 16. Conditional on X, let Y have a Poisson distribution with mean X. Prove that Y has probability generating function E(sY ) = λλ + 1 − sβ.
=+16. Suppose X has gamma density λβxβ−1e−λx/Γ(β) on (0, ∞), where βis not necessarily an integer. Show that X has characteristic function( λλ−it )β and Laplace transform ( λλ+t
=+15. Consider a sequence X1, X2,... of independent, integer-valued random variables with common logarithmic distribution Pr(Xi = k) = − qk k ln(1 − q)for k ≥ 1. Let N be a Poisson random
=+the random variable S is uniformly distributed on [0, 1], and Xi can be interpreted as the ith binary digit of S [192]. In this special case also prove the well-known identity sin θθ = ∞j=1
=+14. Let X1, X2,... be an i.i.d. sequence of Bernoulli random variables with success probability p. Thus, Xi = 1 with probability p, and Xi = 0 with probability 1 − p. Demonstrate that the
=+13. Let Sn = X1 + ··· + Xn be the sum of n independent random variables, each distributed uniformly over the set {1, 2,...,m}. Find the probability generating function of Sn, and use it to
=+12. Let X be a nonnegative integer-valued random variable with probability generating function Q(s). Find the probability generating functions of X + k and kX in terms of Q(s) for any nonnegative
=+Use these facts to show that the t-distribution with 2m degrees of freedom has finite expansionΓ(m + 1/2)√2πmΓ(m) y−∞ 1 + x2 2m−m−1/2 dx= 1 2√2m⎡⎣y√πm−1 j=0Γ(j +
=+11. Consider the integral I(a, p, y) = y−∞1(a + x2)p dx for p > 1 2 and a > 0. As an example of the method of parametric differentiation [29], prove that I(a, p + n, y) = (−1)n p(p +
=+10. In table tennis suppose that player B wins a set with probability p and player A wins a set with probability q = 1 − p. Each set counts 1 point. The winner of a match is the first to
=+9. In the family planning model, suppose the couple has an upper limit m on the number of children they can afford. Hence, they stop whenever they reach their goal of s sons and d daughters or m
=+8. Give a recursive method for computing the second moments E(N2 sd)in the family planning model.
=+7. Consider an urn with b ≥ 1 black balls and w ≥ 0 white balls. Balls are extracted from the urn without replacement until a black ball is encountered. Show that the number of balls Nbw
=+6. A noncentral chi-square random variable X has a χ2 n+2Y distribution conditional on a Poisson random variable Y with mean λ. Show that E(X) = n + 2λ and Var(X)=2n + 8λ.
=+5. Prove that the ln Xn = n i=1 lnUi random variable of Example 2.3.2 follows a −1 2χ2 2n distribution.
=+4. Show that the beta-binomial distribution of Example 2.3.1 has discrete density Pr(Sn = k) = n kΓ(α + β)Γ(α + k)Γ(β + n − k)Γ(α)Γ(β)Γ(α + β + n) .
=+3. Numbers are drawn randomly from the set {1, 2,...,n} until their sum exceeds k for 0 ≤ k ≤ n. Show that the expected number of draws equals ek =1 +1 nk.In particular, en ≈e. (Hint: Show
=+2. In a certain building, p people enter an elevator stationed on the ground floor. There are n floors above the ground floor, and each is an equally likely destination. If the people exit the
=+1. Let X represent the number of fixed points of a random permutation of the set {1,...,n}. Demonstrate that X has the falling factorial moment E[X(X − 1)···(X − k + 1)] = k! EX k = 1
=+23. The Hadamard product C = A ◦ B of two matrices A = (aij ) and B = (bij ) of the same dimensions has entries cij = aij bij . If A and B are nonnegative definite matrices, then show that A ◦
=+22. For n ≥ m, verify the QR decomposition (1.13). (Hints: Write At = (a1,...,am)Q = (q1,...,qn)R = (r1,...,rm).The Gram-Schmidt orthogonalization process applied to the columns of At yields
=+21. Continuing Problem 20, let X be a multivariate normal random vector with mean vector μ and invertible variance matrix Ω. If X has n components, then show that the quadratic form (X −
=+20. Let X1,...,Xn be a sequence of independent standard normal random variables. Prove that χ2 n = X2 1 + ··· + X2 n has a gamma distribution. Calculate the mean and variance of χ2 n.

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