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396Distribution of the Maximum of n Uniform Random VariablesLet X 1, \ldots, X n \sim iid Uniform (0,1). Derive the CDF and PDF of M = \max(X 1, \ldots, X n).概率简单derivation未尝试免费397Reciprocal of a Uniform Random VariableLet X \sim Uniform (0,1). Use the change-of-variables formula to derive the PDF of Y = 1/X. Determine whether E[Y] is finite.概率简单derivation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399Absolute Value of a Standard Normal: The Half-Normal DistributionLet X \sim N(0,1) and define Y = |X|. (a) Derive the PDF of Y using the CDF method (note that Y = |X| is not monotone). (b) Compute E[Y] and Var (Y).概率中等multi part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