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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443Series System Replacement Costs via Competing ExponentialsA machine has two critical components in series: component A with lifetime Exp (3) and component B with lifetime Exp (5), independent of each other. When either fails, the entire machine stops, the failed component is replaced (cost \20 for A, \50 for B), and both components restart fresh (the survivor restarts by memorylessness, the replacement is new). Find the expected replacement cost per unit time in the long run.概率中等数值题未尝试免费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464Delta Method for the Log of a Gamma Sample MeanLet X 1, \ldots, X n be i.i.d.\ Gamma (2, 1) (shape 2, rate 1), so E[X i] = 2 and Var (X i) = 2. Define W n = \ln( X n). **(a)** Using the delta method, determine the asymptotic distribution of n (W n - \ln 2). **(b)** For n = 200, approximate P(W n < 0.6). You may use \ln 2 \approx 0.6931 and \Phi(-1.86) \approx 0.0314.概率困难derivation未尝试免费466Election Poll Margin of Error via the CLTA pollster surveys n = 1 , 600 voters to estimate the proportion p supporting a candidate. Suppose the true proportion is p = 0.5. Using the CLT, find the probability that the sample proportion p falls within 0.02 of the true value, i.e., approximate P(| p - 0.5| < 0.02). You may use \Phi(1.60) \approx 0.9452.概率简单数值题未尝试免费468Aggregate Insurance Claims via the CLTAn insurer has 300 independent policyholders. Each policyholder files a Poisson (3) number of claims per year. Let T = \sum i=1 300 N i be the total number of claims. Using the CLT, approximate P(T > 960). You may use \Phi(2.00) \approx 0.9772.概率中等数值题未尝试免费470Asymptotic Distribution of the Sample MedianLet X 1, \ldots, X n be i.i.d.\ Uniform (0,1) and let M n denote the sample median (the middle order statistic for odd n, or the average of the two middle values for even n). The asymptotic theory of order statistics gives n \,(M n - m) \xrightarrow d N\! (0, 1 4[f(m)] 2 ), where m is the population median and f is the density at m. **(a)** For Uniform (0,1), identify m and f(m), and state the asymptotic variance of n \,M n. **(b)** For n = 400, approximate P(M 400 > 0.54). You may use \Phi(1.60) \approx 0.9452.概率困难derivation未尝试免费472Sample Size for Desired Estimation AccuracyA random variable X has mean = 5 and standard deviation = 2. You observe n independent copies X 1, \ldots, X n and compute X n. Using the CLT, find the smallest n such that P(| X n - 5| > 0.3) < 0.05. You may use \Phi(1.96) \approx 0.975.概率简单数值题未尝试免费