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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