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160Expected and Variance of Distinct Birthday CountAmong n people whose birthdays are independent and uniform on \ 1, \ldots, d\ , let D be the number of distinct birthdays observed. (a) Derive E[D] using indicator random variables. (b) Derive Var (D). You will need P( day j and day k both occupied ) for j \ne k. (c) For n = 100 and d = 365, compute E[D], Var (D), and the expected number of "collision people" n - D (people whose birthday coincides with at least one other person). (d) Is E[n - D] the same as the expected number of collision pairs \binom n 2 /d from the indicator-pair approach? Explain the distinction.概率困难derivation未尝试面试订阅185Coupon Collector: Mean and VarianceA machine dispenses one of n = 4 equally likely prize types per trial. Let T be the number of trials needed to collect all 4 types. Derive both E[T] and Var (T) by decomposing T into independent geometric phases. Express each answer as an exact fraction.概率困难derivation未尝试免费194Variance of the Number of Occupied UrnsFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Let N be the number of nonempty urns. Find Var (N). Give an exact fraction.概率困难数值题未尝试免费248Mean, Variance, and Bimodality of a Gaussian MixtureA random variable X is drawn from a mixture of two normals: with probability p we sample from N(\mu 1, \sigma 1 2) and with probability 1-p from N(\mu 2, \sigma 2 2). (a) Derive E[X] and Var (X) in terms of p, \mu 1, \mu 2, \sigma 1 2, \sigma 2 2. (b) For the symmetric case p = 1/2 and \sigma 1 = \sigma 2 = , show that the mixture PDF is bimodal if and only if |\mu 1 - \mu 2| > 2 . (c) Compute E[X] and Var (X) for p = 1/2, \mu 1 = -2, \mu 2 = 2, \sigma 1 = \sigma 2 = 1, and verify that this mixture is bimodal.概率中等derivation未尝试免费333Robust Law of Total Variance with a Random Number of Coin FlipsYou first draw N \sim Poisson (5). Then, given N = n, you flip a fair coin n times and let X be the number of heads. What is Var (X)? 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339Robust Conditional Variance in the Bivariate NormalLet (X, Y) follow a bivariate normal distribution with E[X] = 0, E[Y] = 0, Var (X) = 1, Var (Y) = \sigma Y 2, and Corr (X,Y) = . Derive Var (Y \mid X = x) and show that it does not depend on x. Evaluate numerically for \sigma Y = 3 and = 0.6. 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344Robust Approximate Variance of a Ratio via the Delta MethodLet X and Y be independent random variables with E[X] = 10, Var (X) = 4, E[Y] = 5, and Var (Y) = 1. Using the delta method (first-order Taylor expansion), derive an approximation for Var (X/Y) and evaluate it numerically. 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350Robust Exact Variance of the Sample Variance for a Normal PopulationLet X 1, \ldots, X n be iid N( , 2) and define the sample variance S 2 = 1 n-1 \sum i=1 n (X i - X ) 2. (a) Identify the distribution of (n-1)S 2 / 2 and use it to derive an exact expression for Var (S 2). (b) Evaluate Var (S 2) when n = 10 and 2 = 3. 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355Beta-Binomial Moments via Adam's and Eve's LawsLet P \sim Beta (2, 3) and, given P = p, let X \sim Binomial (10, p). Using Adam's law (E[X] = E[E[X \mid P]]) and Eve's law ( Var (X) = E[ Var (X \mid P)] + Var (E[X \mid P])), derive E[X] and Var (X).概率困难derivation未尝试免费358Law of Total Variance for a Poisson-Compounded Exponential SumLet N \sim Poisson (3) and, given N = n, let S = X 1 + \cdots + X n where X i \stackrel iid \sim Exp (2) (rate 2). Using the law of total variance, find Var (S).概率中等数值题未尝试免费363Two-Layer Tower with Bernoulli-Switched Exponential RateLet Z \sim Bernoulli (1/2). Given Z = 1, let Y \sim Exp (1); given Z = 0, let Y \sim Exp (2) (rate parametrisation). Given Y = y, let X \sim Poisson (y). Using iterated applications of the tower property and Eve's law, find E[X] and Var (X).概率中等数值题未尝试免费365Three-Level Normal Hierarchy: Iterated Tower and SmoothingConsider a three-level normal hierarchy: Z \sim N(0, 1), then Y \mid Z \sim N(Z, 1), then X \mid Y \sim N(Y, 1). (a) Using iterated expectations, find E[X] and Var (X). (b) Find E[X \mid Z] by applying the tower property: E[X \mid Z] = E[E[X \mid Y] \mid Z]. (c) Verify part (b) by computing Cov (X, Z) and using the joint normality of (X, Z).概率困难derivation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369Three-Layer Poisson-Binomial-Uniform TowerLet U \sim Uniform (0,1). Given U, let N \mid U \sim Poisson (10U). Given (N, U), let X \mid N, U \sim Binomial (N, U). Using iterated tower properties and Eve's law, find E[X] and Var (X).概率困难数值题未尝试免费373Two-Step Tower in an Additive Bernoulli Markov ChainLet X 1 \sim Uniform \ 0, 1\ . Given X 1, let X 2 = X 1 + B 1 where B 1 \sim Bernoulli (1/2) independent of X 1. Given X 2, let X 3 = X 2 + B 2 where B 2 \sim Bernoulli (1/2) independent of everything else. (a) Using the tower property E[X 3 \mid X 1] = E[E[X 3 \mid X 2] \mid X 1], find E[X 3 \mid X 1] and E[X 3]. (b) Using Eve's law, find Var (X 3).概率中等数值题未尝试免费374Compound Poisson Sum: Mean and Variance via Eve's LawLet N \sim Poisson (4) and, given N, let X 1, \ldots, X N be i.i.d.\ with E[X i] = 3 and Var (X i) = 2. Set S = X 1 + \cdots + X N (with S = 0 when N = 0). Using the tower property and Eve's law, find E[S] and Var (S).概率中等数值题未尝试免费375Poisson-Exponential Sum with Shared Rate: Double Tower and Eve's LawLet Z \sim Uniform (1, 3). Given Z = z, let N \mid Z \sim Poisson (z), and given (N, Z), let X 1, \ldots, X N be i.i.d.\ Exp (z) (rate z). Set S = X 1 + \cdots + X N (with S = 0 when N = 0). Using iterated tower properties and Eve's law, find E[S] and Var (S).概率困难derivation未尝试免费