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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408PDF of the Second Smallest ExponentialLet X 1, X 2, X 3, X 4 be independent Exp (1) random variables. Derive the PDF of the second order statistic X (2) .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411Probability Involving the Third Order StatisticLet X 1, X 2, X 3, X 4 be independent Uniform (0,1) random variables. Compute P(X (3) < 0.5), where X (3) is the third smallest.概率简单数值题未尝试免费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413Beta Distribution of the k-th Uniform Order StatisticLet X 1, \ldots, X n be iid Uniform (0,1). Derive that the k-th order statistic X (k) has the Beta (k, n-k+1) distribution.概率中等derivation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416Second Smallest of Four UniformsLet X 1, X 2, X 3, X 4 be independent Uniform (0,1) random variables. Compute E[X (2) ].概率简单数值题未尝试免费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418Expected Value of the Second Smallest ExponentialLet X 1, \ldots, X 5 be independent Exp (1) random variables. Derive E[X (2) ].概率中等derivation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420Variance of the k-th Uniform Order StatisticLet X 1, \ldots, X n be iid Uniform (0,1). Derive a closed-form expression for Var (X (k) ) for 1 \le k \le n.概率困难derivation未尝试面试订阅421CDF of the Minimum of Four UniformsLet X 1, X 2, X 3, X 4 be independent Uniform (0,1) random variables. Derive the CDF and PDF of X (1) = \min(X 1, X 2, X 3, X 4).概率简单derivation未尝试免费422Expected Maximum of Two UniformsLet X 1, X 2 be independent Uniform (0,1) random variables. Compute E[\max(X 1, X 2)].概率简单数值题未尝试免费423Variance of the Range of Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1) and let R = X (n) - X (1) . Derive Var (R) as a function of n.概率中等derivation未尝试免费424Expected Gap Between the Two Largest UniformsLet X 1, \ldots, X 6 be iid Uniform (0,1). Find the expected gap between the largest and second-largest values: E[X (6) - X (5) ].概率中等数值题未尝试免费425Ratio of the Two Smallest Exponential Order StatisticsLet X 1, X 2 be independent Exp (1) random variables with order statistics X (1) \le X (2) . Define U = X (1) / X (2) .概率困难multi part未尝试面试订阅427Conditional Expectation via MemorylessnessLet X \sim Exp (2). Using the memoryless property, compute E[X \mid X > 3].概率简单数值题未尝试免费