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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334Robust Covariance of Overlapping Sums of Independent VariablesLet X, Y, Z be independent random variables with Var (X) = 1, Var (Y) = 2, and Var (Z) = 3. Define U = X + Y and V = Y + Z. Compute Cov (U, V) and Corr (U, V). 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