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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