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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465Berry-Esseen Bound for a Sum of Uniform Random VariablesLet U 1, \ldots, U n be i.i.d.\ Uniform (0,1) and S n = \sum i=1 n U i. The Berry-Esseen theorem states \sup x |P\! ( S n - n/2 n \le x ) - \Phi(x) | \le C\, 3 n , where 2 = Var (U i), = E[|U i - 1/2| 3], and C \le 0.4748. **(a)** Compute = E[|U i - 1/2| 3] exactly. **(b)** Evaluate the Berry-Esseen bound for n = 50. **(c)** How large must n be for the bound to guarantee the CLT error is below 0.01?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467Symmetric Random Walk Displacement via CLTA particle performs a symmetric random walk: at each step i, it moves X i = +1 or X i = -1, each with probability 1/2, independently. After n = 400 steps, the position is S 400 = \sum i=1 400 X i. **(a)** What does the LLN say about S n / n as n ? **(b)** Using the CLT, approximate P(S 400 > 10). You may use \Phi(0.50) \approx 0.6915.概率中等数值题未尝试免费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469Geometric Mean of Random Multipliers via LLN and CLTAn investment grows by a random factor each year: in year i, the portfolio is multiplied by X i, where X i are i.i.d.\ with P(X i = 2) = P(X i = 4) = 1/2. After n years, the annualized growth factor is the geometric mean G n = (\prod i=1 n X i ) 1/n . **(a)** Find \lim n G n almost surely. **(b)** For n = 100, use the CLT to approximate P(G 100 > 3). You may use \ln 2 \approx 0.6931, \ln 3 \approx 1.0986, and \Phi(1.70) \approx 0.9554.概率困难derivation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471Binomial Tail with Continuity CorrectionA fair coin is flipped n = 144 times. Let S be the number of heads. **(a)** Using the CLT (without continuity correction), approximate P(S \ge 80). **(b)** Repeat with the continuity correction. You may use \Phi(1.33) \approx 0.9082 and \Phi(1.25) \approx 0.8944.概率简单数值题未尝试免费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473Probability of Negative Portfolio Return via CLTA portfolio consists of n = 50 stocks with equal weight 1/n. The annual returns R 1, \ldots, R 50 are independent, each with mean = 0.08 (i.e., 8\%) and standard deviation = 0.20. The portfolio return is R = 1 50 \sum i=1 50 R i. **(a)** State what the LLN implies about R as n . **(b)** Using the CLT, approximate P( R < 0). You may use \Phi(2.83) \approx 0.9977.概率中等数值题未尝试免费474Sample Size for a Tail Probability GuaranteeLet X 1, X 2, \ldots be i.i.d.\ Exp (1) (mean 1, variance 1). A system designer requires that the sample mean X n exceeds 1.1 with probability less than 1\%. Using the CLT, find the smallest n satisfying P( X n > 1.1) < 0.01. You may use \Phi(2.33) \approx 0.9901.概率中等数值题未尝试免费475CLT with Estimated Variance via Slutsky's TheoremLet X 1, \ldots, X n be i.i.d.\ with mean and finite variance 2 > 0. Define the sample variance S n 2 = 1 n-1 \sum i=1 n (X i - X n) 2 and the studentized statistic T n = n \,( X n - ) S n . **(a)** Using the LLN and Slutsky's theorem, show that T n \xrightarrow d N(0,1). **(b)** In a study with n = 100 observations, you find X 100 = 12.5 and S 100 = 3.0. Assuming the true mean is \mu 0 = 12, approximate P( X 100 > 12.5) using T n. You may use \Phi(1.67) \approx 0.9525.概率困难derivation未尝试免费590Weighted Offer Stop Rule 5You may inspect up to 3 independent offers. Each offer takes values 0 with probability 1/4, 4 with probability 1/4, 7 with probability 1/4, 12 with probability 1/4. Rejecting an offer and continuing costs 1 point(s), and if you reach the last draw you must accept it. What first-round acceptance threshold is optimal, and what is the resulting expected net payoff?概率困难derivation未尝试免费597Continuation Value Calibration 2A trader may inspect up to 3 independent candidate fills with support [1, 4, 8, 11] and probabilities ['1/4', '1/4', '1/4', '1/4']. Rejecting a fill and continuing costs 1. What first observation becomes just good enough to accept, and what is the overall optimal expected net value?概率简单数值题未尝试免费1439Logarithmic Sequence CorrectionCompute lim n->∞ n 2 [ln(1+1/n) - 1/n].数学困难数值题未尝试面试订阅1445Product-Log CancellationCompute lim x->0 [ (1+x)ln(1+x) - x ] / x 2.数学困难数值题未尝试面试订阅1456Contracting Second-Order Limit 1A sequence obeys x (n+2) = 1/4 x (n+1) + 1/5 x n + 3. If the sequence converges, what must its limit be?数学简单数值题未尝试免费1467Repeating-Block Power Series 2Let a n repeat the block [2, -1] forever. Evaluate sum (n=1) inf a n * (1/4) n.数学中等derivation未尝试免费1469Repeating-Block Power Series 4Let a n repeat the block [1, 2, 0, 1] forever. Evaluate sum (n=1) inf a n * (1/2) n.数学困难derivation未尝试免费1476Clipped Floor Integral 1Evaluate integral 0 1 max(x, 1/5) dx.数学简单derivation未尝试免费