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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368Second Moment of a Scale-Mixed Normal via TowerLet \Theta be drawn uniformly from \ 1, 2, 3\ and, given \Theta = , let X \mid \Theta = \sim N(0, ). Using the tower property, compute E[X 2] and E[X 4].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371Beta-Uniform Prior on Binomial Success ProbabilityLet P \sim Uniform (0,1) and, given P = p, let X \mid P = p \sim Binomial (10, p). Using the tower property, find E[X].概率简单数值题未尝试免费372Expected Maximum of Correlated Bernoullis via Indicator and TowerLet U \sim Uniform (0,1) and, given U, let X and Y be conditionally i.i.d.\ Bernoulli (U). Define M = \max(X, Y). Using the tower property and the indicator representation M = 1 \ X \ge 1 or Y \ge 1\ , find E[M].概率简单数值题未尝试免费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