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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224Compound Poisson Distribution: MGF and MomentsLet N \sim Poisson ( ) and let X 1, X 2, \ldots be iid discrete random variables (independent of N) with PMF P(X i = j) = p j for j = 1, 2, \ldots and MGF M X(t) = E[e tX 1 ]. Define the **compound Poisson** sum S = \sum i=1 N X i (with S = 0 when N = 0). (a) Derive the MGF of S. Show that M S(t) = \exp\!\big( (M X(t) - 1)\big). (b) Use the MGF to derive E[S] and Var (S). Express your answers in terms of , E[X 1], and Var (X 1). (c) Alternatively, derive E[S] and Var (S) using the tower property (law of total expectation) and the Eve's law (law of total variance), conditioning on N. (d) **Application:** An insurance company receives claims at a Poisson rate of = 10 per day. Each claim size is \1000 with probability 0.6 or \5000 with probability 0.4, independently. Find E[S] and Var (S) for the total daily claims S, and compute the standard deviation.概率困难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226Mean and Variance of the Continuous Uniform DistributionLet X \sim Uniform (a, b). Starting from the PDF f(x) = 1 b-a for x \in [a, b], derive E[X] and Var (X).概率简单derivation未尝试免费227Memorylessness of the Exponential DistributionLet X \sim Exponential ( ). Prove that P(X > s + t \mid X > s) = P(X > t) for all s, t 0. Then show the exponential is the only continuous distribution with this property.概率简单derivation未尝试免费228Moment Generating Function of the Gamma DistributionLet X \sim Gamma ( , ) with PDF f(x) = \Gamma( ) x -1 e - x for x > 0. Derive the moment generating function M X(t) = E[e tX ] for t < . Then use M X(t) to compute E[X] and Var (X).概率中等derivation未尝试免费229Standard Normal Tail and Interval ProbabilitiesLet Z \sim N(0,1). Using \Phi(1) \approx 0.8413 and \Phi(2) \approx 0.9772, compute: (a) P(|Z| > 2) and (b) P(1 < Z < 2). Explain each step using symmetry of the normal distribution.概率中等数值题未尝试免费230Beta Distribution: Derivation, Moments, and ConnectionsConsider the Beta distribution with PDF f(x) = \Gamma( + ) \Gamma( )\Gamma( ) x -1 (1-x) -1 for x \in (0,1). (a) Verify that f integrates to 1 by showing \int 0 1 x -1 (1-x) -1 \,dx = \Gamma( )\Gamma( ) \Gamma( + ) using the integral representation of the Gamma function. (b) Derive E[X] = + and Var (X) = ( + ) 2( + +1) . (c) Show that Beta (1,1) reduces to Uniform (0,1). (d) If Y 1 \sim Gamma ( , 1) and Y 2 \sim Gamma ( , 1) are independent, explain (without full proof) why Y 1 Y 1+Y 2 \sim Beta ( , ).概率困难derivation未尝试免费231CDF and Tail Probability of the Exponential DistributionLet X \sim Exponential ( ) with PDF f(x) = e - x for x 0. (a) Derive the CDF F(x) = P(X x). (b) If = 1 2 , compute P(X > 3).概率简单数值题未尝试免费