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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410Joint Density and Covariance of Two Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1). Consider the order statistics X (i) and X (j) with 1 \le i < j \le n.概率困难multi part未尝试面试订阅412Expected Minimum of Five UniformsLet X 1, \ldots, X 5 be independent Uniform (0,1) random variables. Find E[X (1) ], the expected value of the minimum.概率简单数值题未尝试免费414Renyi Representation of Exponential Order-Statistic SpacingsLet X 1, \ldots, X n be iid Exp ( ) and let X (1) \le \cdots \le X (n) be the order statistics. Define the normalized spacings D k = (n-k+1)(X (k) - X (k-1) ) for k = 1, \ldots, n, where X (0) = 0.概率困难multi part未尝试面试订阅415Distribution of the Mid-Range for Uniform SamplesLet X 1, \ldots, X n be iid Uniform (0,1) with n \ge 2. The mid-range is defined as M = X (1) + X (n) 2 . Using the joint density of (X (1) , X (n) ), derive the PDF of M.概率困难derivation未尝试面试订阅417Probability That the Range Exceeds One-HalfLet X 1, X 2, X 3 be independent Uniform (0,1) random variables. The range is R = X (3) - X (1) . Compute P(R > \tfrac 1 2 ).概率中等数值题未尝试免费419Conditional Distribution of the Minimum Given the MaximumLet X 1, \ldots, X n be iid Uniform (0,1) with n \ge 3. Let X (1) and X (n) denote the minimum and maximum.概率困难multi part未尝试面试订阅