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305Robust Unique Choices and Unique NeighborsThere are n people in a line, and each independently and uniformly picks an integer from \ 1, 2, \dots, k\ . A person is called unique if no other person picked the same number. (a) Using indicator variables, find E[U], the expected number of unique people. (b) A unique neighbor pair is a pair of adjacent people (i, i+1) who are both unique. Find E[N], the expected number of unique neighbor pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费315Robust Singleton Coupons After Random DrawsA collector draws m coupons independently and uniformly at random from n types. A coupon type is called a singleton if it appears exactly once among the m draws. Find the expected number of singleton types. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费323Robust Overlap of Two Random SubsetsLet S and T be two subsets of \ 1, 2, \dots, n\ , each chosen independently and uniformly at random from all \binom n k subsets of size k (where 1 \le k \le n). Find the expected size of their intersection |S \cap T|. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation未尝试免费325Robust Comparable Pairs in Random PointsLet X 1, X 2, \dots, X n be independent and uniformly distributed on [0,1] d (the d-dimensional unit hypercube). Two points X i and X j are called comparable if one dominates the other coordinatewise, i.e., either X i \le X j in every coordinate or X j \le X i in every coordinate. Find the expected number of comparable pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费333Robust Law of Total Variance with a Random Number of Coin FlipsYou first draw N \sim Poisson (5). Then, given N = n, you flip a fair coin n times and let X be the number of heads. What is Var (X)? Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费335Robust Variance of the Sample Mean Under Sampling Without ReplacementAn urn contains N balls numbered 1, 2, \dots, N. You draw n balls without replacement and let X = \tfrac 1 n \sum i=1 n X i, where X i is the number on the i-th draw. Derive Var ( X ) in terms of N and n, and evaluate it for N = 10, n = 4. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费339Robust Conditional Variance in the Bivariate NormalLet (X, Y) follow a bivariate normal distribution with E[X] = 0, E[Y] = 0, Var (X) = 1, Var (Y) = \sigma Y 2, and Corr (X,Y) = . Derive Var (Y \mid X = x) and show that it does not depend on x. Evaluate numerically for \sigma Y = 3 and = 0.6. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费344Robust Approximate Variance of a Ratio via the Delta MethodLet X and Y be independent random variables with E[X] = 10, Var (X) = 4, E[Y] = 5, and Var (Y) = 1. Using the delta method (first-order Taylor expansion), derive an approximation for Var (X/Y) and evaluate it numerically. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费350Robust Exact Variance of the Sample Variance for a Normal PopulationLet X 1, \ldots, X n be iid N( , 2) and define the sample variance S 2 = 1 n-1 \sum i=1 n (X i - X ) 2. (a) Identify the distribution of (n-1)S 2 / 2 and use it to derive an exact expression for Var (S 2). (b) Evaluate Var (S 2) when n = 10 and 2 = 3. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费355Beta-Binomial Moments via Adam's and Eve's LawsLet P \sim Beta (2, 3) and, given P = p, let X \sim Binomial (10, p). Using Adam's law (E[X] = E[E[X \mid P]]) and Eve's law ( Var (X) = E[ Var (X \mid P)] + Var (E[X \mid P])), derive E[X] and Var (X).概率困难derivation未尝试免费358Law of Total Variance for a Poisson-Compounded Exponential SumLet N \sim Poisson (3) and, given N = n, let S = X 1 + \cdots + X n where X i \stackrel iid \sim Exp (2) (rate 2). Using the law of total variance, find Var (S).概率中等数值题未尝试免费364Tower Property Verification in a Gaussian Markov ChainLet (X, Y, Z) be mean-zero jointly normal with Var (X) = Var (Y) = Var (Z) = 1, Corr (X,Y) = 1/2, Corr (Y,Z) = 1/3, and Corr (X,Z) = 1/6. (This makes X - Y - Z a Gaussian Markov chain: X \perp\!\!\perp Z \mid Y.) (a) Compute E[X \mid Z] directly using the bivariate normal regression formula. (b) Compute E[E[X \mid Y] \mid Z] by first finding E[X \mid Y], then taking its conditional expectation given Z. (c) Verify that both answers agree, illustrating the tower property E[X \mid Z] = E[E[X \mid Y] \mid Z] when (Z) \subseteq (Y) is replaced by the Markov condition X \perp\!\!\perp Z \mid Y.概率困难derivation未尝试免费365Three-Level Normal Hierarchy: Iterated Tower and SmoothingConsider a three-level normal hierarchy: Z \sim N(0, 1), then Y \mid Z \sim N(Z, 1), then X \mid Y \sim N(Y, 1). (a) Using iterated expectations, find E[X] and Var (X). (b) Find E[X \mid Z] by applying the tower property: E[X \mid Z] = E[E[X \mid Y] \mid Z]. (c) Verify part (b) by computing Cov (X, Z) and using the joint normality of (X, Z).概率困难derivation未尝试免费369Three-Layer Poisson-Binomial-Uniform TowerLet U \sim Uniform (0,1). Given U, let N \mid U \sim Poisson (10U). Given (N, U), let X \mid N, U \sim Binomial (N, U). Using iterated tower properties and Eve's law, find E[X] and Var (X).概率困难数值题未尝试免费375Poisson-Exponential Sum with Shared Rate: Double Tower and Eve's LawLet Z \sim Uniform (1, 3). Given Z = z, let N \mid Z \sim Poisson (z), and given (N, Z), let X 1, \ldots, X N be i.i.d.\ Exp (z) (rate z). Set S = X 1 + \cdots + X N (with S = 0 when N = 0). Using iterated tower properties and Eve's law, find E[S] and Var (S).概率困难derivation未尝试免费380Distribution of the Ratio of Two Independent ExponentialsLet X and Y be independent Exp (1) random variables. Define R = X/Y. (a) Using the transformation (R, S) = (X/Y,\, Y), compute the joint density f R,S via the Jacobian and then marginalize over S to find the PDF of R. (b) Identify f R as a named distribution and verify by computing P(R \le 1) using a symmetry argument.概率困难derivation未尝试免费383Reciprocal Transform of an Exponential VariableLet X \sim Exp (1). Define Y = \dfrac 1 1 + X . (a) Use the change-of-variables formula to derive the PDF of Y. (b) Compute E[Y].概率中等derivation未尝试免费385Box-Muller Transform: From Uniforms to Independent NormalsLet U 1, U 2 be independent Uniform (0,1) random variables. Define Z 1 = -2\ln U 1 \,\cos(2 U 2), \qquad Z 2 = -2\ln U 1 \,\sin(2 U 2). (a) Compute the Jacobian of the inverse transformation from (Z 1, Z 2) back to (U 1, U 2). (b) Show that Z 1 and Z 2 are independent N(0,1) random variables.概率困难derivation未尝试免费388Odds Transformation of a Beta Variable Yields Beta PrimeLet X \sim Beta (a, b) with a, b > 0. Use the change-of-variables formula to derive the PDF of Y = \dfrac X 1 - X and identify the resulting distribution.概率中等derivation未尝试免费389Ratio of Independent Gammas Yields a Beta DistributionLet X \sim Gamma ( , 1) and Y \sim Gamma ( , 1) be independent. Using the transformation (W, S) = \bigl(X/(X+Y),\; X+Y\bigr): (a) Compute the Jacobian of the inverse map. (b) Derive the joint density f W,S and marginalize to show W \sim Beta ( , ). (c) Show that W and S are independent.概率困难derivation未尝试免费