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109Expected Wait for the First HeartA standard 52-card deck is shuffled uniformly at random. Cards are turned over one at a time from the top. Let X be the position of the first heart. Find E[X].概率中等derivation未尝试免费161Expected Near-Birthday PairsForty people have independent uniform birthdays on a 365-day circular calendar. What is the expected number of unordered pairs whose birthdays are either the same day or one day apart on the circle?概率简单数值题未尝试免费162Triple-Collision Expectation Above OneIn a 365-day uniform birthday model, what is the smallest n for which the expected number of unordered triples sharing an exact birthday is at least 1?概率简单数值题未尝试免费163Expected Number of Exact Triple CollisionsFor n independent uniform birthdays on an m-day calendar, what is the expected number of unordered triples of people that share an exact birthday?概率中等derivation未尝试面试订阅165Expected Number of People Who Share Someone's BirthdayFor n independent uniform birthdays on a 365-day calendar, what is the expected number of people whose birthday is shared by at least one other person?概率困难derivation未尝试面试订阅166Expected Cross-Team Birthday MatchesTeam A has 12 people and team B has 18 people. Birthdays are independent and uniform over 365 days. What is the expected number of cross-team exact birthday matches (one person from A and one from B)?概率简单数值题未尝试免费168Pair Expectation Under a Nonuniform CalendarBirthdays fall on days 1 through m with probabilities p1,...,pm, not necessarily uniformly. For n independent people, what is the expected number of unordered matching pairs?概率中等derivation未尝试面试订阅169Expected Pairs Within Two DaysOn a 365-day circular calendar, what is the expected number of unordered pairs among n independent uniform birthdays whose birthdays are at circular distance at most 2 days?概率困难derivation未尝试面试订阅170Collision Risk for the Next Arrival Given Distinct Existing BirthdaysSuppose n existing birthdays are all distinct on a 365-day calendar. A new person's birthday is then drawn uniformly and independently. What is the probability the new arrival creates an exact collision?概率困难derivation未尝试面试订阅171Expected Pair Count With One Holiday Twice as LikelyOn a 366-point calendar, one special holiday has probability 2/366 and each of the other 364 days has probability 1/366. For n=30 independent birthdays, what is the expected number of unordered matching pairs?概率简单数值题未尝试免费172Poisson Approximation Threshold for Near CollisionsUse the pair-Poisson approximation for birthdays on a 365-day circular calendar where a pair counts as a collision if the birthdays are the same day or one day apart. What is the smallest n for which the approximated collision probability exceeds 50%?概率简单数值题未尝试免费173Expected Number of Singleton BirthdaysFor n independent uniform birthdays on a 365-day calendar, what is the expected number of people whose birthday is unique in the sample?概率中等derivation未尝试面试订阅174Expected Number of Days With Exactly Two BirthdaysFor n independent uniform birthdays on a 365-day calendar, what is the expected number of calendar days that receive exactly two birthdays?概率中等derivation未尝试面试订阅175Expected Number of Days With at Least Three BirthdaysFor n independent uniform birthdays on a 365-day calendar, what is the expected number of calendar days that receive at least three birthdays?概率困难derivation未尝试面试订阅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