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385Box-Muller Transform: From Uniforms to Independent NormalsLet U 1, U 2 be independent Uniform (0,1) random variables. Define Z 1 = -2\ln U 1 \,\cos(2 U 2), \qquad Z 2 = -2\ln U 1 \,\sin(2 U 2). (a) Compute the Jacobian of the inverse transformation from (Z 1, Z 2) back to (U 1, U 2). (b) Show that Z 1 and Z 2 are independent N(0,1) random variables.概率困难derivation未尝试免费386Affine Transformation of a Normal Random VariableLet X \sim N( , 2) with a 0 and b \in R . Using the Jacobian formula, show that Y = aX + b is normally distributed and state its parameters.概率简单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388Odds Transformation of a Beta Variable Yields Beta PrimeLet X \sim Beta (a, b) with a, b > 0. Use the change-of-variables formula to derive the PDF of Y = \dfrac X 1 - X and identify the resulting distribution.概率中等derivation未尝试免费389Ratio of Independent Gammas Yields a Beta DistributionLet X \sim Gamma ( , 1) and Y \sim Gamma ( , 1) be independent. Using the transformation (W, S) = \bigl(X/(X+Y),\; X+Y\bigr): (a) Compute the Jacobian of the inverse map. (b) Derive the joint density f W,S and marginalize to show W \sim Beta ( , ). (c) Show that W and S are independent.概率困难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391Square Root of an Exponential Random VariableLet X \sim Exp (1). Use the change-of-variables formula to derive the PDF of Y = X and identify the resulting distribution.概率简单derivation未尝试免费392Tangent of a Uniform Variable Yields the Cauchy DistributionLet X \sim Uniform (- /2, /2). Derive the PDF of Y = \tan(X) using the change-of-variables formula and identify the resulting distribution.概率中等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394Ratio of Independent Standard Normals Is CauchyLet X 1, X 2 \sim iid N(0,1). Using the transformation (Y, V) = (X 1/X 2,\, X 2): (a) Derive the joint density f Y,V (y,v). (b) Integrate out V to obtain the marginal PDF of Y = X 1/X 2 and identify the distribution.概率困难derivation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404Expected Range of Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1). The range is defined as R = X (n) - X (1) . Derive a closed-form expression for E[R] as a function of n.概率中等derivation未尝试免费405Joint Distribution of Extremes and the RangeLet X 1, \ldots, X n be iid Uniform (0,1). Let X (1) = \min i X i and X (n) = \max i X i.概率困难multi part未尝试面试订阅406Second Order Statistic from Five UniformsLet X 1, \ldots, X 5 be independent Uniform (0,1) random variables and let X (2) denote the second smallest. Find E[X (2) ].概率简单数值题未尝试免费