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188Capped Weak CompositionsSeven indistinguishable jobs are split across four labeled servers. How many occupancy vectors are possible if no server is allowed to receive more than three jobs?概率中等数值题未尝试免费189Probability of a Heavily Loaded UrnSix distinguishable balls are thrown independently and uniformly at random into 4 distinguishable urns. What is the probability that at least one urn contains 3 or more balls? Give an exact fraction.概率困难数值题未尝试免费190Expected Number of Odd-Load BinsEight labeled balls are independently assigned to five labeled bins. What is the expected number of bins that end up with an odd load?概率困难数值题未尝试免费191Two Tagged Balls Stay IsolatedBalls 1 and 2 are tagged. Six labeled balls are independently assigned to four labeled bins. What is the probability that ball 1 and ball 2 land in different bins and each of those two bins contains no other ball?概率简单数值题未尝试免费192Exact Occupancy Profile 2-2-1-0Five labeled tasks are independently assigned to four labeled queues. How many assignments produce queue loads that, after sorting, equal (2,2,1,0)?概率中等数值题未尝试免费193Conditional Expected Occupancy of a Nonempty UrnFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Given that urn 1 is nonempty, what is the expected number of balls in urn 1? Give an exact fraction.概率中等数值题未尝试免费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195Expected Time to First Collision in Six UrnsBalls are thrown one at a time, each landing independently and uniformly at random into one of 6 urns. Let T be the index of the first ball that lands in an already-occupied urn (so T \ge 2). Derive E[T] and give an exact fraction.概率困难derivation未尝试免费196Strictly Descending Loads Across Three Named BinsSeven labeled packets are independently assigned to three labeled bins A, B, C. How many assignments produce loads satisfying load(A) > load(B) > load(C)?概率简单数值题未尝试免费197Expected Number of Heavy BinsEight labeled balls are independently assigned to four labeled bins. What is the expected number of bins whose load is at least three?概率简单数值题未尝试免费199Expected Maximum Urn OccupancyFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Let M = \max(X 1, X 2, X 3) be the maximum number of balls in any single urn. Find E[M]. Give an exact fraction.概率中等数值题未尝试免费200Full Distribution of Empty Urns via Stirling NumbersSix distinguishable balls are thrown independently and uniformly at random into 5 distinguishable urns. Let E denote the number of empty urns. Derive the probability mass function P(E = k) for every possible value of k, expressing each probability as an exact fraction.概率困难derivation未尝试免费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