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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387Probability Integral Transform (Inverse CDF Method)Let F X be a continuous, strictly increasing CDF and U \sim Uniform (0,1). Prove that Y = F X -1 (U) has CDF F X. Conversely, show that if X has CDF F X, then F X(X) \sim Uniform (0,1).概率简单derivation未尝试免费390Linear Transformation of a Multivariate Normal via MGFLet X \sim N(\boldsymbol , \boldsymbol \Sigma ) be a p-dimensional normal random vector, and let A be a fixed m p matrix. Using the moment-generating function, prove that Z = A X is multivariate normal and determine its mean vector and covariance matrix.概率困难derivation未尝试免费393Sum of Squares of Two Standard NormalsLet X 1, X 2 \sim iid N(0,1). Define R = X 1 2 + X 2 2. (a) Switching to polar coordinates (X 1, X 2) = (r\cos , r\sin ), derive the joint density of (R, \Theta) where R = X 1 2 + X 2 2 and \Theta = \arctan(X 2/X 1). (b) Marginalize over \Theta to find the PDF of R and identify its distribution.概率中等multi part未尝试免费395Exponential of a Normal: The Log-Normal DistributionLet X \sim N( , 2) and define Y = e X. (a) Derive the PDF of Y using the change-of-variables formula. (b) Using the MGF of the normal distribution, compute E[Y] and Var (Y). (c) Show that the median of Y is e and explain why E[Y] > median (Y) when > 0.概率困难multi part未尝试免费398Additivity of Chi-Squared Distributions via MGFLet X \sim \chi 2(m) and Y \sim \chi 2(n) be independent. Using moment-generating functions, prove that X + Y \sim \chi 2(m + n).概率中等derivation未尝试免费400Deriving the Fisher F-Distribution from Chi-Squared VariablesLet X \sim \chi 2(m) and Y \sim \chi 2(n) be independent. Define F = X/m Y/n . (a) Using the transformation (F, W) = \bigl( nX mY ,\; Y\bigr), compute the Jacobian and derive the joint density f F,W . (b) Integrate out W to obtain the marginal PDF of F and verify it matches the F(m, n) distribution. (c) Show that E[F] = \dfrac n n-2 for n > 2.概率困难multi part未尝试免费402Distribution of the Minimum of Exponential Random VariablesLet X 1, \ldots, X n be independent Exp ( ) random variables. Derive the distribution of X (1) = \min(X 1, \ldots, X n).概率简单derivation未尝试免费407Variance of the Maximum of Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1). Derive a closed-form expression for Var (X (n) ) as a function of n.概率中等derivation未尝试免费409Expected Spacing Between Consecutive Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1) and let X (0) = 0, X (n+1) = 1. Show that E[X (k+1) - X (k) ] = 1 n+1 for every k = 0, 1, \ldots, n, and compute this value for n = 4.概率中等数值题未尝试免费