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159Near-Birthday Problem: Birthdays Within One DayFourteen people have birthdays chosen independently and uniformly on a circular calendar of 365 days (day 1 is adjacent to day 365). Two people have a **near-match** if their birthdays differ by at most 1 day (i.e., they land on the same day or on consecutive days). Let M be the number of unordered near-match pairs. (a) Compute E[M]. (b) Using a Poisson approximation for the probability that M \ge 1, estimate P( at least one near-match ). (c) Contrast with the standard birthday problem: for n = 14 people, what is P( at least one exact match )?概率中等数值题未尝试免费160Expected and Variance of Distinct Birthday CountAmong n people whose birthdays are independent and uniform on \ 1, \ldots, d\ , let D be the number of distinct birthdays observed. (a) Derive E[D] using indicator random variables. (b) Derive Var (D). You will need P( day j and day k both occupied ) for j \ne k. (c) For n = 100 and d = 365, compute E[D], Var (D), and the expected number of "collision people" n - D (people whose birthday coincides with at least one other person). (d) Is E[n - D] the same as the expected number of collision pairs \binom n 2 /d from the indicator-pair approach? Explain the distinction.概率困难derivation未尝试面试订阅164Exactly One Pair Among Four BirthdaysFour people have independent uniform birthdays on a 365-day calendar. What is the probability that there is exactly one matching pair and no larger collision?概率困难derivation未尝试面试订阅176Server A Empty but Server B ActiveSix labeled jobs are independently routed to four labeled servers A, B, C, D with equal probability. How many assignments leave server A empty but give server B at least one job?概率简单数值题未尝试免费177Odd Load on a Tagged ServerFive labeled packets are independently assigned to three labeled servers. In how many assignments does server A receive an odd number of packets?概率简单数值题未尝试免费182Expected Occupied Servers After a Tagged AssignmentA tagged job is known to have been routed to server A. Then five additional labeled jobs are independently routed uniformly to the four servers A, B, C, D. What is the expected total number of occupied servers?概率中等数值题未尝试免费183Collision Probability in the Occupancy ModelFive distinguishable balls are thrown independently and uniformly at random into 12 distinguishable urns. What is the probability that at least two balls land in the same urn? Give an exact fraction.概率中等数值题未尝试免费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