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202Poisson Approximation to the BinomialA factory produces 500 chips per day and each chip independently has a defect probability of 0.01. Using the Poisson approximation to the binomial, estimate the probability that exactly 3 chips are defective on a given day.概率简单数值题未尝试免费204Negative Binomial Variance from Geometric SummationLet X be the number of independent Bernoulli(p) trials needed to accumulate r successes. Express X as a sum of independent geometric random variables and use this to find Var (X).概率中等derivation未尝试免费205Hypergeometric Mean and Variance via Indicator VariablesAn urn contains 20 balls: 8 red and 12 blue. You draw 5 balls without replacement. Let X be the number of red balls drawn. Define indicator variables X i = 1 \ ball i is red \ for each draw i = 1, \ldots, 5. (a) Use linearity of expectation to find E[X]. (b) Compute Cov (X i, X j) for i \ne j and use it to derive Var (X). (c) Verify that your variance formula reduces to the binomial variance np(1-p) when N with K/N p held fixed.概率困难derivation未尝试免费206Memorylessness of the Geometric DistributionLet X \sim Geometric (p) count the number of independent Bernoulli(p) trials until the first success (so P(X = k) = (1-p) k-1 p for k = 1, 2, \ldots). (a) Derive a closed-form expression for P(X > n). (b) Prove the memorylessness property: for all positive integers m, n, P(X > m + n \mid X > m) = P(X > n). (c) Is there any other discrete distribution on \ 1, 2, 3, \ldots\ that is memoryless? Justify briefly.概率简单derivation未尝试免费207Binomial Moments via the Moment Generating FunctionLet X \sim Binomial (n, p). (a) Derive the moment generating function M X(t) = E[e tX ] in closed form. (b) By differentiating M X(t) and evaluating at t = 0, find E[X] and E[X 2], and hence Var (X).概率中等derivation未尝试免费208Closure of Poisson under Independent SummationLet X \sim Poisson ( ) and Y \sim Poisson ( ) be independent. (a) Derive the MGF of a Poisson ( ) random variable. (b) Using MGFs, prove that X + Y \sim Poisson ( + ). (c) A call centre receives calls from two independent sources at rates = 3 and = 7 per hour. What is the probability of receiving exactly 8 calls in the next hour?概率中等derivation未尝试免费209Negative Binomial PMF from First PrinciplesLet X be the number of independent Bernoulli(p) trials needed to accumulate exactly r successes. (a) By considering the structure of the sequence of outcomes up to trial X, derive the PMF P(X = k) for k = r, r+1, r+2, \ldots (b) Verify that your PMF sums to 1 using the negative binomial series (1-x) -r = \sum j=0 \binom r+j-1 j x j for |x| < 1. (c) Compute P(X = 7) when r = 3 and p = 1/2.概率中等derivation未尝试免费210Multinomial Covariance and Conditional DistributionA fair die is rolled n = 60 times. Let X i be the number of times face i appears, for i = 1, \ldots, 6, so (X 1, \ldots, X 6) \sim Multinomial (60,\, 1/6, \ldots, 1/6). (a) Using indicator variables, compute Cov (X 1, X 2). (b) Find the correlation (X 1, X 2). (c) Determine the conditional distribution of (X 2, X 3, X 4, X 5, X 6) given X 1 = 12. What is E[X 2 \mid X 1 = 12]?概率困难derivation未尝试免费212Geometric Moments via the Probability Generating FunctionLet X \sim Geometric (p) with P(X = k) = (1-p) k-1 p for k = 1, 2, 3, \ldots (a) Derive the probability generating function G X(s) = E[s X] in closed form for |s| < 1 1-p . (b) Using G X, compute E[X] and Var (X). Recall: E[X] = G X'(1) and Var (X) = G X''(1) + G X'(1) - [G X'(1)] 2.概率简单derivation未尝试免费213Conditional Distribution of a Binomial Sum ComponentLet X \sim Binomial (m, p) and Y \sim Binomial (n, p) be independent. (a) What is the distribution of S = X + Y? State and justify. (b) Derive the conditional PMF P(X = k \mid S = s) for valid k. (c) Identify this conditional distribution by name and parameters. Interpret the result: why does p disappear from the conditional distribution? (d) Verify with a numerical example: m = 10, n = 15, p = 0.4. Compute P(X = 3 \mid S = 8).概率中等derivation未尝试免费214Poisson Thinning and Independence of Split StreamsLet N \sim Poisson ( ). Each of the N events is independently classified as type 1 with probability p and type 2 with probability 1 - p. Let N 1 and N 2 denote the counts of type 1 and type 2 events, respectively. (a) Derive the marginal distribution of N 1. (b) Derive the joint PMF P(N 1 = j, N 2 = k) and show that N 1 and N 2 are independent. (c) A website receives page views at rate = 200 per hour. Each visitor independently converts (makes a purchase) with probability p = 0.03. Find the probability of exactly 4 conversions in an hour, and the probability of at least 1 conversion given at most 210 total page views.概率困难derivation未尝试免费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217Negative Binomial as a Poisson–Gamma MixtureLet \Lambda \sim Gamma (r, ) with density f \Lambda( ) = r \Gamma(r) r-1 e - for > 0, and let X \mid \Lambda = \sim Poisson ( ). (a) Write down P(X = k \mid \Lambda = ) and compute the marginal PMF P(X = k) by integrating over \Lambda. (b) Show that P(X = k) = \binom k + r - 1 k p k (1-p) r where p = 1 1+ , and identify the distribution. (c) Use the mixture representation to find E[X] and Var (X) via the tower property (law of total expectation and law of total variance), without computing the PMF. (d) Verify numerically: r = 3, = 4. Compute P(X = 2) and E[X].概率中等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