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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285Robust Monochromatic Cliques in a Random Edge-ColoringEach edge of the complete graph K n is independently colored red or blue with equal probability 1 2 . For a fixed integer k \ge 2, find the expected number of monochromatic k-cliques (complete subgraphs on k vertices whose edges are all the same color). Express your answer in terms of n and k. 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286Robust Ascents in a Random PermutationLet be a uniformly random permutation of \ 1, 2, \dots, n\ . An ascent at position i (for 1 \le i \le n-1) is a position where (i) < (i+1). Find the expected number of ascents. 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288Robust Isolated Vertices in a Random GraphIn the Erdos-Renyi random graph model G(n,p), each of the \binom n 2 possible edges among n labeled vertices is included independently with probability p. A vertex is isolated if it has no edges. Find the expected number of isolated vertices. 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derivation未尝试免费295Robust Cycles in a Random PermutationLet be a uniformly random permutation of \ 1, 2, \dots, n\ . Find the expected number of cycles in the cycle decomposition of . Express your answer as a familiar function of n. 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300Robust Common Edges of Two Random Spanning TreesLet T 1 and T 2 be two independent uniformly random spanning trees of the complete graph K n (each drawn uniformly at random from all n n-2 labeled spanning trees, independently of the other). Find the expected number of edges that belong to both T 1 and T 2. 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676Inventory-Spread State Repeat 1A simulator records the pair (inventory mod 9, spread bucket among 5 buckets) once per minute. What is the minimum number of recorded minutes that guarantees two records have exactly the same pair?脑筋急转弯简单数值题未尝试免费677Two-Step Recurrence Repeat 2A deterministic recurrence over mod 13 is fully determined by the ordered pair (x t, x (t+1)). How many consecutive ordered pairs force a repeated state pair?脑筋急转弯简单brainteaser未尝试免费