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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376Distribution of the Cube of a Uniform Random VariableLet X \sim Uniform (0,1). Using the CDF method, derive the PDF of Y = X 3.概率简单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379Distribution and Mean of the Maximum of Two ExponentialsLet X and Y be independent Exp (1) random variables. Define M = \max(X, Y). (a) Derive the PDF of M. (b) Compute E[M].概率中等数值题未尝试免费380Distribution of the Ratio of Two Independent ExponentialsLet X and Y be independent Exp (1) random variables. Define R = X/Y. (a) Using the transformation (R, S) = (X/Y,\, Y), compute the joint density f R,S via the Jacobian and then marginalize over S to find the PDF of R. (b) Identify f R as a named distribution and verify by computing P(R \le 1) using a symmetry argument.概率困难derivation未尝试免费381Negative Log of a Uniform Yields an ExponentialLet X \sim Uniform (0,1). Using the CDF method, derive the PDF of Y = -\ln X and identify the resulting distribution.概率简单derivation未尝试免费382Square of a Standard Normal Is Chi-Squared(1)Let X \sim N(0,1). Using the CDF method, derive the PDF of Y = X 2 and identify the resulting distribution.概率中等derivation未尝试免费383Reciprocal Transform of an Exponential VariableLet X \sim Exp (1). Define Y = \dfrac 1 1 + X . (a) Use the change-of-variables formula to derive the PDF of Y. (b) Compute E[Y].概率中等derivation未尝试免费384Distribution of the Product of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the transformation (W, V) = (XY,\, Y), derive the PDF of W = XY.概率中等derivation未尝试免费