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216Mean and Variance of the Discrete Uniform DistributionLet X be uniformly distributed on \ 1, 2, \ldots, n\ , so P(X = k) = 1/n for each k. (a) Derive E[X] in closed form. (b) Derive E[X 2] using the identity \sum k=1 n k 2 = n(n+1)(2n+1) 6 . (c) Obtain Var (X). (d) Evaluate numerically for a fair six-sided die (n = 6).概率简单derivation未尝试免费218Coupon Collector's Problem via Geometric Waiting TimesA cereal box contains one of n distinct coupon types, each equally likely. You buy boxes one at a time, independently. Let T be the number of boxes needed to collect all n types. (a) Define T i as the number of additional boxes needed to go from i-1 distinct types to i distinct types. What is the distribution of T i? State its parameter. (b) Express T in terms of T 1, T 2, \ldots, T n and use linearity of expectation to derive E[T]. (c) Show that E[T] = n H n where H n = \sum k=1 n 1/k is the n-th harmonic number. (d) Compute E[T] for n = 10. How many boxes on average? (e) Derive Var (T) using the independence of T 1, \ldots, T n.概率中等derivation未尝试免费221Bernoulli Distribution: Moments and MGFLet X \sim Bernoulli (p), so P(X=1)=p and P(X=0)=1-p where 0 < p < 1. (a) Compute E[X] and E[X 2] directly from the PMF. Hence find Var (X). (b) Derive the moment-generating function M X(t) = E[e tX ] and verify that M X'(0) = E[X] and M X''(0) = E[X 2]. (c) Show that Var (X) = p(1-p) \le 1/4 for all p \in (0,1), and identify the value of p that maximizes the variance. (d) If S = \sum i=1 n X i where X 1, \ldots, X n are iid Bernoulli (p), use the MGF to show S \sim Binomial (n, p).概率简单derivation未尝试免费222Convolution of Two Independent Poisson Random VariablesLet X \sim Poisson (2) and Y \sim Poisson (3), independent. (a) Compute P(X + Y = 3) using the convolution formula P(X+Y=k) = \sum j=0 k P(X=j)P(Y=k-j). (b) Verify your answer using the fact that X + Y \sim Poisson (5). (c) Compute P(X = 1 \mid X + Y = 3).概率简单数值题未尝试免费223Zero-Truncated Poisson DistributionThe **zero-truncated Poisson** distribution arises when a Poisson process is observed conditional on at least one event occurring. Let Y \sim Poisson ( ) with > 0, and define X = (Y \mid Y \ge 1). (a) Derive the PMF of X: show that P(X = k) = k k!(e - 1) for k = 1, 2, 3, \ldots, and verify it sums to 1. (b) Compute E[X] by relating it to E[Y] via the truncation. Specifically, show that E[X] = 1 - e - . (c) Derive Var (X) using the identity Var (X) = E[X 2] - (E[X]) 2. Show that Var (X) = (1 + ) 1 - e - - 2 (1 - e - ) 2 . (d) Evaluate P(X = 1), E[X], and Var (X) numerically for = 0.5.概率中等derivation未尝试免费225Minimum of Independent Geometric Random VariablesLet X 1, X 2, \ldots, X n be independent, each X i \sim Geometric (p i) with P(X i = k) = (1-p i) k-1 p i for k = 1, 2, \ldots (the "number of trials until first success" convention). Define M = \min(X 1, \ldots, X n). (a) Show that P(M > k) = \prod i=1 n (1-p i) k for k = 0, 1, 2, \ldots (b) Prove that M \sim Geometric \! (1 - \prod i=1 n (1-p i) ). State the PMF of M explicitly. (c) For the iid case p i = p for all i: express E[M] and Var (M) in terms of n and p, and verify that E[M] 1 as n . (d) **Application:** Five independent traders each attempt to fill an order on any given day with probability 0.3. What is the expected number of days until the first fill occurs? What is the probability that no fill occurs in the first 3 days? (e) Show that P(X j = M \mid M = m) depends on j (when p i are not all equal) and compute this probability for n = 2, p 1 = 0.3, p 2 = 0.5, m = 2.概率困难derivation未尝试免费236Median of the Weibull DistributionThe Weibull distribution with shape k > 0 and scale > 0 has CDF F(x) = 1 - e -(x/ ) k , \quad x 0. (a) Derive a closed-form expression for the median of this distribution. (b) Compute the median when k = 2 and = 3.概率简单数值题未尝试免费247Erlang Distribution as Poisson Process Waiting TimeArrivals follow a Poisson process with rate > 0. Let T k denote the time of the k-th arrival. (a) Using the relationship P(T k > t) = P(N(t) < k) where N(t) \sim Poisson ( t), write the CDF of T k and differentiate to obtain its PDF. (b) Identify the distribution by name and state its mean and variance. (c) For = 1 and k = 3, compute P(T 3 > 2) as an exact numerical value.概率简单数值题未尝试免费249Order Statistics of the Uniform Distribution: Min, Max, and RangeLet X 1, X 2, \ldots, X n be iid Uniform (0, 1) random variables. Denote the minimum by X (1) and the maximum by X (n) . (a) Derive the PDF of X (1) and the PDF of X (n) . (b) Compute E[X (1) ] and E[X (n) ]. (c) The range is W = X (n) - X (1) . Compute E[W] for n = 5.概率中等数值题未尝试免费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638Normalized Product Process 3Let Y 1, Y 2, ... be iid positive random variables taking values [1, 4] with probabilities ['3/4', '1/4'], and let F n = sigma(Y 1,...,Y n). Define M n = (Y 1...Y n)/(7/4) n. Is (M n) a martingale?概率中等derivation未尝试免费641Martingale Diagnosis Counterexample 1M n = S n/(n+1), where S n = X 1+...+X n for iid symmetric ±1 increments. Is (M n) a martingale?概率简单derivation未尝试免费647Predictable-Transform Martingale 2M n = sum (k=1) n (1+S (k-1) 2) * X k, where X k are iid symmetric ±1. Is (M n) a martingale with respect to the natural filtration?概率简单数值题未尝试免费656Fair Corridor Exit Time 1A symmetric random walk starts at 1 and stops when it first hits 0 or 6. What is the expected stopping time?概率中等数值题未尝试免费673Scaled-Step Exit Time 3A fair random walk starts at 3 and moves by +3 or -3 with equal probability each step. It stops when it first hits -9 or 9. What is the expected stopping time?概率中等数值题未尝试免费690Repeated State in a 20-State Engine 5A deterministic trading engine is described by a state pair (inventory mod 4, cash-balance bucket mod 5). Starting from any state and recording one state per observation, how many observations are enough to guarantee that some state is seen at least twice?脑筋急转弯中等数值题未尝试面试订阅692Subset Size and Residue Collision 7Among any 36 distinct subsets of 1,2,3,4,5,6 , show that two subsets must have the same size and the same sum of elements modulo 5.脑筋急转弯中等derivation未尝试面试订阅693Smallest Sample for a Two-Label Integer Collision 8Each integer n is labeled by two features: its remainder modulo 5, and the parity of floor(n/5). What is the smallest N such that any choice of N integers must contain two with exactly the same pair of labels?脑筋急转弯中等数值题未尝试免费892Two-Segment Demand Total 2A forecast splits annual package deliveries into Segment A with 18000 entities at 1.4 events each and Segment B with 6000 entities at 3.1 events each. What is the combined annual estimate?脑筋急转弯中等数值题未尝试免费894Two-Segment Demand Total 4A forecast splits annual support chats into Segment A with 15000 entities at 2.5 events each and Segment B with 3000 entities at 6 events each. What is the combined annual estimate?脑筋急转弯困难数值题未尝试面试订阅