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447The Memoryless BusA bus arrives at a stop at an Exp (1/10) random time (mean 10 minutes). You have already been waiting for 5 minutes. What is the expected additional waiting time?概率简单数值题未尝试免费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477Mean Return Time and the Stationary DistributionA Markov chain on \ 1, 2, 3\ has transition matrix P = \begin pmatrix 0 & \tfrac 1 2 & \tfrac 1 2 \\ \tfrac 1 4 & \tfrac 1 2 & \tfrac 1 4 \\ \tfrac 1 3 & \tfrac 1 3 & \tfrac 1 3 \end pmatrix . **(a)** Find the stationary distribution . **(b)** Using the relationship between the stationary distribution and mean return times, compute the expected number of steps to return to state 1 starting from state 1.概率简单数值题未尝试免费486Three-State First Passage with Deterministic ReturnA Markov chain on \ 0, 1, 2\ has transition probabilities: p(0,1) = 1, \quad p(1,0) = \tfrac 2 5 , \quad p(1,2) = \tfrac 3 5 , \quad p(2,2) = 1. Compute E[T 2 \mid X 0 = 0], where T 2 = \inf\ n \ge 0 : X n = 2\ .概率简单数值题未尝试免费496First Passage in a Three-State Chain with Self-LoopA Markov chain on \ 0, 1, 2\ has transition matrix P = \begin pmatrix 1 & 0 & 0 \\ \tfrac 1 5 & \tfrac 2 5 & \tfrac 2 5 \\ 0 & \tfrac 3 4 & \tfrac 1 4 \end pmatrix . State 0 is absorbing. Compute E[T 0 \mid X 0 = 1], where T 0 = \inf\ n \ge 1 : X n = 0\ .概率简单数值题未尝试免费