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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429Geometric Number of Geometric TrialsA gambler plays a sequence of rounds. In each round, he flips a biased coin with P( heads ) = p repeatedly until he gets heads; the number of flips in that round is Geom (p). The number of rounds he plays is itself Geom (q) (independent of the coin flips), where 0 < q < 1. Let S be the total number of coin flips across all rounds. Using the memoryless property of the geometric distribution, show that S \sim Geom (pq) and compute E[S].概率中等derivation未尝试免费430Characterization of Memorylessness and the Residual Life ParadoxPart (a): Let X be a continuous, positive random variable satisfying P(X > s + t \mid X > s) = P(X > t) for all s, t 0. Prove that X must be exponentially distributed. Part (b): A lightbulb's lifetime L has CDF F(t) = 1 - 1 2 e -t - 1 2 e -3t for t 0 (a mixture of Exp (1) and Exp (3)). You arrive at a uniformly random time and observe the bulb currently in use. Let R be the residual lifetime of that bulb. Show that E[R] > E[L] and compute both values. Explain why memorylessness breaks down and causes this paradox.概率困难derivation未尝试面试订阅431Geometric Survival Past a ThresholdLet X \sim Geom (1/4) (number of trials until first success). Using the memoryless property of the geometric distribution, compute (i) E[X \mid X > 5] and (ii) P(X > 8 \mid X > 5).概率简单数值题未尝试免费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433Conditional Variance of a Surviving ExponentialLet X \sim Exp ( ). Using the memoryless property, find Var (X \mid X > t) for t > 0. Does conditioning on survival change the variance compared to Var (X)? Evaluate numerically for = 5 and t = 2.概率中等数值题未尝试免费