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324Robust Monochromatic Edges in a Random ColoringEach vertex of the complete graph K n is independently colored red or blue, each with probability \tfrac 1 2 . An edge is monochromatic if both its endpoints have the same color. Find the expected number of monochromatic edges. 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367Variance of a Geometric-Stopped Exponential SumLet N \sim Geometric (1/2) (so P(N = k) = (1/2) k for k = 1, 2, \ldots) and, given N, let X 1, \ldots, X N be i.i.d.\ Exp (1). Set S = X 1 + \cdots + X N. Using the law of total expectation and Eve's law, find E[S] and Var (S).概率中等数值题未尝试免费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373Two-Step Tower in an Additive Bernoulli Markov ChainLet X 1 \sim Uniform \ 0, 1\ . Given X 1, let X 2 = X 1 + B 1 where B 1 \sim Bernoulli (1/2) independent of X 1. Given X 2, let X 3 = X 2 + B 2 where B 2 \sim Bernoulli (1/2) independent of everything else. (a) Using the tower property E[X 3 \mid X 1] = E[E[X 3 \mid X 2] \mid X 1], find E[X 3 \mid X 1] and E[X 3]. (b) Using Eve's law, find Var (X 3).概率中等数值题未尝试免费374Compound Poisson Sum: Mean and Variance via Eve's LawLet N \sim Poisson (4) and, given N, let X 1, \ldots, X N be i.i.d.\ with E[X i] = 3 and Var (X i) = 2. Set S = X 1 + \cdots + X N (with S = 0 when N = 0). Using the tower property and Eve's law, find E[S] and Var (S).概率中等数值题未尝试免费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377Distribution of the Exponential of a Uniform via JacobianLet X \sim Uniform (0,1). Use the change-of-variables (Jacobian) formula to find the PDF of Y = e X.概率简单derivation未尝试免费378Distribution of the Sum of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the convolution formula, derive the PDF of Z = X + Y.概率中等derivation未尝试免费