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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