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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435Uniqueness of Geometric MemorylessnessPart (a): Let N be a positive-integer-valued random variable satisfying P(N > m + n \mid N > m) = P(N > n) for all m, n \in Z 0 . Prove that N must follow a geometric distribution. Part (b): For N \sim Geom (p), compute E[N 2 \mid N > k] using memorylessness and verify that Var (N \mid N > k) = Var (N).概率困难derivation未尝试面试订阅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445Memorylessness Breaks for Exponential MixturesLet X have the mixture density f(x) = 1 2 e -x + 5 2 e -5x for x 0 (a 50 - 50 mixture of Exp (1) and Exp (5)). (a) Compute P(X > s + t \mid X > s) as a function of s and t, and show it depends on s (i.e., the memoryless property fails). (b) Evaluate P(X > 2 \mid X > 1) and compare with P(X > 1). (c) Interpret: given that X has survived past s, how does the conditional distribution change as s increases?概率困难multi part未尝试面试订阅450Head Start in an Exponential RaceLet X \sim Exp ( ) and Y \sim Exp ( ) be independent. Player A finishes at time X and player B finishes at time Y + c where c > 0 is a head start for player A (player B starts c time units later). (a) Derive P(X < Y + c) — the probability that A finishes before B. (b) Show that as c 0, the result recovers the standard competing-exponentials formula. (c) Evaluate for = 3, = 2, c = 1 and interpret how the head start affects A's winning probability.概率困难multi part未尝试面试订阅451Normal Approximation to Coin-Flip CountsA fair coin is flipped n = 400 times independently. Let S denote the total number of heads. Using the Central Limit Theorem, approximate P(190 \le S \le 210). You may use the fact that \Phi(1) \approx 0.8413, where \Phi is the standard normal CDF.概率简单数值题未尝试免费452Sample Mean of Exponential Service TimesA server processes 100 independent requests. Each request takes Exp (1) time (mean 1 second). Let T = 1 100 \sum i=1 100 T i be the average processing time. **(a)** State what the Law of Large Numbers guarantees about T as n . **(b)** Using the CLT, approximate P( T > 1.2). You may use \Phi(2) \approx 0.9772.概率简单数值题未尝试免费453Call Center Overflow via Poisson CLTA call center receives calls according to a Poisson process with rate = 4 calls per minute. The center operates for an 8-hour shift (480 minutes). The center can handle at most 2000 calls per shift before service quality degrades. Using a suitable normal approximation, estimate the probability that the total number of calls in a single shift exceeds 2000. You may use the following: \Phi(1.83) \approx 0.9664.概率中等数值题未尝试免费454Berry-Esseen Bound for a Skewed Bernoulli SumLet X 1, X 2, \ldots, X n be i.i.d.\ Bernoulli (p) with p = 0.01 and n = 10 , 000. Define S n = \sum i=1 n X i. **(a)** Using the CLT, approximate P(S n \le 80). **(b)** The Berry-Esseen theorem states that \sup x |P(Z n \le x) - \Phi(x)| \le C\, 3 n , where Z n = (S n - n )/( n ), = E[|X 1 - | 3], and C \le 0.4748. Compute the Berry-Esseen bound on the approximation error in part (a). You may use \Phi(-2) \approx 0.0228.概率中等derivation未尝试免费455Geometric Mean of Random Gains via LLN and CLTLet X 1, X 2, \ldots be i.i.d.\ with P(X i = 1) = \tfrac 1 2 and P(X i = 0) = \tfrac 1 2 . Define the geometric-mean-like quantity Y n = (\prod i=1 n (1 + X i) ) 1/n . **(a)** Find \lim n Y n almost surely. **(b)** For n = 200, use the CLT to approximate P(Y 200 > 1.45). You may use: \ln 2 \approx 0.6931, \Phi(1.02) \approx 0.8461.概率困难derivation未尝试免费456Defective Items in a Production BatchA factory produces items independently, each defective with probability p = 0.03. A batch of n = 500 items is inspected. Let D denote the number of defective items in the batch. Using the Central Limit Theorem, approximate P(D \le 20). You may use \Phi(1.30) \approx 0.9032.概率简单数值题未尝试免费457Sum of Uniform Random Variables Exceeding a ThresholdLet U 1, U 2, \ldots, U 60 be independent Uniform (0,1) random variables, and define S = \sum i=1 60 U i. Using the CLT, approximate P(S > 35). You may use \Phi(2.24) \approx 0.9875.概率中等数值题未尝试免费458Empirical Frequency Accuracy via the CLTA biased die shows a six with probability p = 1/3. You roll it n = 900 times independently and record p = ( number of sixes )/n. **(a)** State what the Law of Large Numbers guarantees about p as n . **(b)** Using the CLT, approximate P(| p - 1/3| < 0.02). You may use \Phi(1.27) \approx 0.8980.概率中等数值题未尝试免费459Continuity Correction in the Normal ApproximationLet S \sim Bin (200, 0.45). Use the CLT with a continuity correction to approximate P(S = 85). Recall that for a discrete integer-valued random variable, P(S = k) \approx \Phi\! ( k + 0.5 - ) - \Phi\! ( k - 0.5 - ). You may use: \Phi(-0.64) \approx 0.2611, \Phi(-0.78) \approx 0.2177.概率中等数值题未尝试免费460Delta Method for Square Root of the Sample MeanLet X 1, \ldots, X n be i.i.d.\ Exp ( ) with = 4 (so E[X i] = 1/4, Var (X i) = 1/16). Define T n = X n . **(a)** Using the delta method, find the asymptotic distribution of n \,(T n - ) where = E[X i]. **(b)** For n = 256, approximate P(T 256 > 0.525). You may use \Phi(1.60) \approx 0.9452.概率困难derivation未尝试免费461Sum of Fair Dice via the CLTYou roll 100 independent fair six-sided dice. Let S denote the sum of all outcomes. Using the CLT, approximate P(340 \le S \le 380). You may use \Phi(0.58) \approx 0.7190 and \Phi(1.75) \approx 0.9599.概率简单数值题未尝试免费462Estimating Pi by Monte Carlo and the Law of Large NumbersTo estimate , you draw n = 10 , 000 points (X i, Y i) independently and uniformly on the unit square [0,1] 2. Define Z i = 1 (X i 2 + Y i 2 \le 1), and let = 4 Z where Z = 1 n \sum i=1 n Z i. **(a)** Explain why E[ ] = and why almost surely. **(b)** Using the CLT, find an approximate 95\% confidence interval for given that the observed Z = 0.7854. You may use \Phi(1.96) \approx 0.975 and \approx 3.1416.概率简单数值题未尝试免费463Inventory Stockout Probability via Poisson CLTA warehouse stocks 240 units of a product each week. Weekly demand follows a Poisson (225) distribution. Using a normal approximation, find the probability that demand exceeds supply in a given week. You may use \Phi(1.00) \approx 0.8413.概率中等数值题未尝试免费