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219Distribution of the Maximum of Independent Geometric Random VariablesLet X 1, X 2, \ldots, X n be independent Geometric (p) random variables with P(X i = k) = (1-p) k-1 p for k = 1, 2, \ldots Define M = \max(X 1, \ldots, X n). (a) Show that P(M \le m) = [1 - (1-p) m] n for m = 1, 2, \ldots (b) Derive P(M = m) from the CDF. (c) Express E[M] as an infinite series using the tail-sum formula E[M] = \sum m=0 P(M > m). Simplify to: E[M] = \sum m=0 [1 - (1 - (1-p) m) n ]. (d) For the special case n = 2, p = 1/2, compute P(M = 1), P(M = 2), P(M = 3) and verify they sum to nearly 1. Compute E[M] exactly by evaluating the series. (e) For general n and small p, argue heuristically that E[M] \approx \ln n p by comparing to the continuous exponential analogue.概率困难derivation未尝试免费220Poisson Limit of the Binomial via Characteristic FunctionsLet X n \sim Binomial (n, /n) for fixed > 0 and n > . (a) Write down the characteristic function \varphi X n (t) = E[e itX n ] in closed form. (b) Show that \lim n \varphi X n (t) = e (e it - 1) for every t \in R . (c) Identify the limiting characteristic function and state the convergence-in-distribution conclusion. (d) Justify why pointwise convergence of characteristic functions implies convergence in distribution (cite the relevant theorem). (e) For = 5, n = 100: compute P(X n = 3) using both the exact Binomial PMF and the Poisson approximation, and find the relative error.概率困难derivation未尝试免费287Robust Same-Rank Pairs in a Poker HandYou are dealt a 5-card hand uniformly at random from a standard 52-card deck. A same-rank pair is an unordered pair of cards in your hand that share the same rank (e.g., two Kings). Find the expected number of same-rank pairs in your hand. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费296Robust Heads-Tails Transitions in Coin FlipsYou flip a fair coin n times independently. A transition at position i (for 1 \le i \le n-1) occurs when flip i and flip i+1 differ (one is heads and the other tails). Find the expected number of transitions. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单derivation未尝试免费297Robust Distinct Birthdays in a GroupA group of n people each have a birthday chosen independently and uniformly at random from 365 days. A day is represented if at least one person in the group has that birthday. Find the expected number of represented days. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单derivation未尝试免费302Robust Matching Colors in a LineThere are n people standing in a line. Each person independently and uniformly picks one of three colors: red, green, or blue. What is the expected number of adjacent pairs (i, i+1) who chose the same color? Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费315Robust Singleton Coupons After Random DrawsA collector draws m coupons independently and uniformly at random from n types. A coupon type is called a singleton if it appears exactly once among the m draws. Find the expected number of singleton types. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费322Robust Adjacent Matches in Dice RollsRoll a fair six-sided die n times independently, producing a sequence D 1, D 2, \dots, D n. An adjacent match occurs at position i (for 1 \le i \le n - 1) if D i = D i+1 . Find the expected number of adjacent matches. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费334Robust Covariance of Overlapping Sums of Independent VariablesLet X, Y, Z be independent random variables with Var (X) = 1, Var (Y) = 2, and Var (Z) = 3. Define U = X + Y and V = Y + Z. Compute Cov (U, V) and Corr (U, V). Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费337Robust Variance of a Difference of Independent VariablesLet X and Y be independent random variables with Var (X) = 4 and Var (Y) = 9. A student claims that SD (X - Y) = SD (X) - SD (Y) = 2 - 3 = -1. Find the correct value of Var (X - Y) and SD (X - Y), and explain the student's error. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费338Robust Variance of a Product of Two Independent Uniform VariablesLet X and Y be independent, each uniformly distributed on [0, 1]. Compute Var (XY). Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费343Robust Covariance of Multinomial CountsA fair six-sided die is rolled 60 times independently. Let N 1 be the number of times face 1 appears and N 2 the number of times face 2 appears. (a) Find Cov (N 1, N 2). (b) Use your answer to compute Var (N 1 + N 2) and verify it by recognizing the distribution of N 1 + N 2. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费349Robust Variance of a Random Sum (Wald's Variance Identity)A shop receives N customer orders per day, where N \sim Poisson (8). Each order has an independent random dollar amount X i with E[X i] = 50 and Var (X i) = 400. Let S = X 1 + X 2 + \cdots + X N be the total daily revenue. Using the law of total variance, derive a formula for Var (S) and evaluate it. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费352Wald's Equation with a Geometric Number of TermsYou roll a fair die repeatedly until you get a 6. Each non-six roll scores the value shown; a roll of 6 scores nothing and ends the game. Let S be your total score. Using Wald's equation, find E[S].概率简单数值题未尝试免费353Second Moment of a Random Sum via the Tower PropertyLet N \sim Poisson (4) and, given N = n, let S = X 1 + \cdots + X n where X i \stackrel iid \sim Uniform (0,1). Use the tower property and the identity E[S 2 \mid N] = Var (S \mid N) + (E[S \mid N]) 2 to find E[S 2].概率中等数值题未尝试免费356Tower Property with a Three-Level Discrete LatentA random variable K is drawn uniformly from \ 1, 2, 3\ . Given K = k, the random variable X \sim Exp (k) (rate k, so E[X \mid K = k] = 1/k). Find E[X].概率简单数值题未尝试免费361Random Number of Coin Flips via Tower PropertyA fair die is rolled to obtain D \sim Uniform \ 1,2,3,4,5,6\ . Then D independent fair coins are flipped and X equals the total number of heads. Using the tower property, find E[X].概率简单数值题未尝试免费362Two-Stage Binomial Draw via Iterated ExpectationLet N be drawn uniformly from \ 1, 2, 3, 4\ , and given N = n, let X \sim Binomial (n, 1/3). Find E[X].概率简单数值题未尝试免费366Product Moment via Tower Conditioning on One FactorLet Y \sim Exp (1) and, given Y = y, let X \mid Y = y \sim Uniform (0, y). Using the tower property, compute E[XY].概率简单数值题未尝试免费367Variance of a Geometric-Stopped Exponential SumLet N \sim Geometric (1/2) (so P(N = k) = (1/2) k for k = 1, 2, \ldots) and, given N, let X 1, \ldots, X N be i.i.d.\ Exp (1). Set S = X 1 + \cdots + X N. Using the law of total expectation and Eve's law, find E[S] and Var (S).概率中等数值题未尝试免费