第 1 / 1 页
非代码面试题
显示 2 / 2 道匹配题目
答题状态：未尝试未正确已正确
ID题目领域难度题型进度权限
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