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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432Asymmetric Penalties in an Exponential RaceTwo independent alarms go off at Exp (4) and Exp (6) times respectively. If alarm 1 fires first you pay \3; if alarm 2 fires first you pay \5. After the first alarm fires, the remaining alarm is reset (memoryless restart) and you pay an additional \1 when it fires. Find the expected total payment.概率中等数值题未尝试免费434Second Failure in a Memoryless Component ArrayA system has 4 independent components, each with lifetime Exp (2). When a component fails, it is removed and the remaining components continue operating. By memorylessness, surviving components' residual lifetimes are still Exp (2). Find the expected time until the second component fails.概率中等数值题未尝试免费438Machine Replacements via Memoryless MinimumA factory runs 3 identical machines with independent lifetimes Exp (1). When any machine fails, it is instantly replaced with a new identical machine. All non-failed machines continue running (their residual lifetimes remain Exp (1) by memorylessness). Find the expected number of machine replacements in the time interval [0, 10].概率中等数值题未尝试免费442Constant Hazard Rate from MemorylessnessA device's lifetime X has survival function F (t) = P(X > t) and hazard rate h(t) = f(t)/ F (t). Show that the memoryless property P(X > s + t \mid X > s) = P(X > t) implies h(t) = (a constant) for all t 0, and conversely that a constant hazard rate implies the memoryless property. Conclude that X \sim Exp ( ).概率中等derivation未尝试免费