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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337Robust Variance of a Difference of Independent VariablesLet X and Y be independent random variables with Var (X) = 4 and Var (Y) = 9. A student claims that SD (X - Y) = SD (X) - SD (Y) = 2 - 3 = -1. Find the correct value of Var (X - Y) and SD (X - Y), and explain the student's error. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费338Robust Variance of a Product of Two Independent Uniform VariablesLet X and Y be independent, each uniformly distributed on [0, 1]. Compute Var (XY). 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343Robust Covariance of Multinomial CountsA fair six-sided die is rolled 60 times independently. Let N 1 be the number of times face 1 appears and N 2 the number of times face 2 appears. (a) Find Cov (N 1, N 2). (b) Use your answer to compute Var (N 1 + N 2) and verify it by recognizing the distribution of N 1 + N 2. 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349Robust Variance of a Random Sum (Wald's Variance Identity)A shop receives N customer orders per day, where N \sim Poisson (8). Each order has an independent random dollar amount X i with E[X i] = 50 and Var (X i) = 400. Let S = X 1 + X 2 + \cdots + X N be the total daily revenue. Using the law of total variance, derive a formula for Var (S) and evaluate it. 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352Wald's Equation with a Geometric Number of TermsYou roll a fair die repeatedly until you get a 6. Each non-six roll scores the value shown; a roll of 6 scores nothing and ends the game. Let S be your total score. Using Wald's equation, find E[S].概率简单数值题未尝试免费353Second Moment of a Random Sum via the Tower PropertyLet N \sim Poisson (4) and, given N = n, let S = X 1 + \cdots + X n where X i \stackrel iid \sim Uniform (0,1). Use the tower property and the identity E[S 2 \mid N] = Var (S \mid N) + (E[S \mid N]) 2 to find E[S 2].概率中等数值题未尝试免费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356Tower Property with a Three-Level Discrete LatentA random variable K is drawn uniformly from \ 1, 2, 3\ . Given K = k, the random variable X \sim Exp (k) (rate k, so E[X \mid K = k] = 1/k). Find E[X].概率简单数值题未尝试免费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359Tower Property with a Continuous Mixing ParameterLet U \sim Uniform (0,1) and, given U = u, let X \sim Geometric (u) (number of trials until first success, so P(X = k \mid U = u) = (1-u) k-1 u for k = 1, 2, \ldots). Find E[X] using the tower property.概率中等数值题未尝试免费361Random Number of Coin Flips via Tower PropertyA fair die is rolled to obtain D \sim Uniform \ 1,2,3,4,5,6\ . Then D independent fair coins are flipped and X equals the total number of heads. Using the tower property, find E[X].概率简单数值题未尝试免费362Two-Stage Binomial Draw via Iterated ExpectationLet N be drawn uniformly from \ 1, 2, 3, 4\ , and given N = n, let X \sim Binomial (n, 1/3). Find E[X].概率简单数值题未尝试免费363Two-Layer Tower with Bernoulli-Switched Exponential RateLet Z \sim Bernoulli (1/2). Given Z = 1, let Y \sim Exp (1); given Z = 0, let Y \sim Exp (2) (rate parametrisation). Given Y = y, let X \sim Poisson (y). Using iterated applications of the tower property and Eve's law, find E[X] and Var (X).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366Product Moment via Tower Conditioning on One FactorLet Y \sim Exp (1) and, given Y = y, let X \mid Y = y \sim Uniform (0, y). Using the tower property, compute E[XY].概率简单数值题未尝试免费