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215Distribution of Dice Sums via Probability Generating FunctionsLet X 1, X 2, \ldots, X n be iid rolls of a fair d-sided die, so each X i is uniform on \ 1, 2, \ldots, d\ . Let S n = X 1 + X 2 + \cdots + X n. (a) Derive the PGF G X 1 (s) = E[s X 1 ] in closed form. (b) Write the PGF of S n and use it to derive E[S n] and Var (S n). (c) For n = 3 fair six-sided dice (d = 6), use the PGF to find P(S 3 = 10). (d) Explain how the coefficient-extraction approach relates to the classical stars-and-bars counting with inclusion-exclusion for this problem.概率困难derivation未尝试免费427Conditional Expectation via MemorylessnessLet X \sim Exp (2). Using the memoryless property, compute E[X \mid X > 3].概率简单数值题未尝试免费429Geometric Number of Geometric TrialsA gambler plays a sequence of rounds. In each round, he flips a biased coin with P( heads ) = p repeatedly until he gets heads; the number of flips in that round is Geom (p). The number of rounds he plays is itself Geom (q) (independent of the coin flips), where 0 < q < 1. Let S be the total number of coin flips across all rounds. Using the memoryless property of the geometric distribution, show that S \sim Geom (pq) and compute E[S].概率中等derivation未尝试免费431Geometric Survival Past a ThresholdLet X \sim Geom (1/4) (number of trials until first success). Using the memoryless property of the geometric distribution, compute (i) E[X \mid X > 5] and (ii) P(X > 8 \mid X > 5).概率简单数值题未尝试免费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436Direct Application of Exponential MemorylessnessA radioactive atom has a lifetime X \sim Exp (1/2). Given that the atom has survived past time t = 3, what is the probability it survives past time t = 7?概率简单数值题未尝试免费437Fresh Start After a Losing StreakYou flip a coin with P( heads ) = 1/3 until you get heads. Given that the first 8 flips were all tails, what is the expected total number of additional flips needed (starting from flip 9)?概率简单数值题未尝试免费441Minimum of Three Identical ExponentialsLet X 1, X 2, X 3 be independent, each Exp (4). Find the distribution of M = \min(X 1, X 2, X 3) and compute E[M]. Then verify: given that M > 2, use memorylessness to find E[M \mid M > 2].概率简单数值题未尝试免费446Geometric Coupon Collector's Conditional SurvivalYou roll a fair die repeatedly until you roll a 6. Let N be the number of rolls needed. Given that you have not yet rolled a 6 in the first 5 rolls, what is the probability that you will need more than 10 rolls total?概率简单数值题未尝试免费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449Memoryless Message Relay ChainA message must traverse a chain of relay nodes to reach its destination. Each node independently takes Geom (1/3) attempts to successfully forward the message to the next node. However, on each attempt, there is an independent probability 1/5 that the node permanently fails, destroying the message. If the node fails, the message is lost. If the chain has 2 nodes, find: (i) The probability the message reaches the destination (traverses both nodes). (ii) The expected total number of attempts across both nodes, given the message reaches the destination.概率中等数值题未尝试免费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476Hitting Time on a Four-State ChainConsider a Markov chain on states \ 0, 1, 2, 3\ with transition probabilities: p(0,1) = 1, \quad p(1,0) = \tfrac 1 3 , \quad p(1,2) = \tfrac 2 3 , \quad p(2,1) = \tfrac 1 2 , \quad p(2,3) = \tfrac 1 2 , \quad p(3,3) = 1. Starting from state 0, compute the expected number of steps to reach state 3 for the first time.概率简单数值题未尝试免费481Hitting Time with Skip TransitionsA Markov chain on \ 0, 1, 2, 3, 4\ has the following transition rules. From state i with 1 \le i \le 3: move to i-1 with probability \tfrac 1 2 , and move to i+1 with probability \tfrac 1 2 . From state 4: jump to state 2 with probability 1. State 0 is absorbing. Compute E[T 0 \mid X 0 = 2], the expected number of steps to reach state 0 starting from state 2.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