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086The Birthday ParadoxAssume birthdays are uniformly distributed over 365 days (ignore leap years). (a) What is the exact probability that in a room of 23 people, at least two share a birthday? (b) Derive a simple approximation for the number of people n needed so that the probability of a shared birthday exceeds 1/2. (c) A trading desk has 50 traders. A manager claims that having two traders share a birthday is 'a remarkable coincidence.' Is the manager correct? Compute the probability and comment.概率简单数值题未尝试免费153Birthday Collision Probability and Exponential ApproximationFifty people are in a room. Each birthday is independent and uniform on \ 1, 2, \ldots, 365\ . (a) Write the exact probability that at least two share a birthday. (b) Derive a useful upper bound on P( all distinct ) using the inequality 1-x \le e -x and simplify. How does the approximation compare to the exact value?概率中等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344Robust Approximate Variance of a Ratio via the Delta MethodLet X and Y be independent random variables with E[X] = 10, Var (X) = 4, E[Y] = 5, and Var (Y) = 1. Using the delta method (first-order Taylor expansion), derive an approximation for Var (X/Y) and evaluate it numerically. 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454Berry-Esseen Bound for a Skewed Bernoulli SumLet X 1, X 2, \ldots, X n be i.i.d.\ Bernoulli (p) with p = 0.01 and n = 10 , 000. Define S n = \sum i=1 n X i. **(a)** Using the CLT, approximate P(S n \le 80). **(b)** The Berry-Esseen theorem states that \sup x |P(Z n \le x) - \Phi(x)| \le C\, 3 n , where Z n = (S n - n )/( n ), = E[|X 1 - | 3], and C \le 0.4748. Compute the Berry-Esseen bound on the approximation error in part (a). You may use \Phi(-2) \approx 0.0228.概率中等derivation未尝试免费455Geometric Mean of Random Gains via LLN and CLTLet X 1, X 2, \ldots be i.i.d.\ with P(X i = 1) = \tfrac 1 2 and P(X i = 0) = \tfrac 1 2 . Define the geometric-mean-like quantity Y n = (\prod i=1 n (1 + X i) ) 1/n . **(a)** Find \lim n Y n almost surely. **(b)** For n = 200, use the CLT to approximate P(Y 200 > 1.45). You may use: \ln 2 \approx 0.6931, \Phi(1.02) \approx 0.8461.概率困难derivation未尝试免费459Continuity Correction in the Normal ApproximationLet S \sim Bin (200, 0.45). Use the CLT with a continuity correction to approximate P(S = 85). Recall that for a discrete integer-valued random variable, P(S = k) \approx \Phi\! ( k + 0.5 - ) - \Phi\! ( k - 0.5 - ). You may use: \Phi(-0.64) \approx 0.2611, \Phi(-0.78) \approx 0.2177.概率中等数值题未尝试免费460Delta Method for Square Root of the Sample MeanLet X 1, \ldots, X n be i.i.d.\ Exp ( ) with = 4 (so E[X i] = 1/4, Var (X i) = 1/16). Define T n = X n . **(a)** Using the delta method, find the asymptotic distribution of n \,(T n - ) where = E[X i]. **(b)** For n = 256, approximate P(T 256 > 0.525). You may use \Phi(1.60) \approx 0.9452.概率困难derivation未尝试免费464Delta Method for the Log of a Gamma Sample MeanLet X 1, \ldots, X n be i.i.d.\ Gamma (2, 1) (shape 2, rate 1), so E[X i] = 2 and Var (X i) = 2. Define W n = \ln( X n). **(a)** Using the delta method, determine the asymptotic distribution of n (W n - \ln 2). **(b)** For n = 200, approximate P(W n < 0.6). You may use \ln 2 \approx 0.6931 and \Phi(-1.86) \approx 0.0314.概率困难derivation未尝试免费465Berry-Esseen Bound for a Sum of Uniform Random VariablesLet U 1, \ldots, U n be i.i.d.\ Uniform (0,1) and S n = \sum i=1 n U i. The Berry-Esseen theorem states \sup x |P\! ( S n - n/2 n \le x ) - \Phi(x) | \le C\, 3 n , where 2 = Var (U i), = E[|U i - 1/2| 3], and C \le 0.4748. **(a)** Compute = E[|U i - 1/2| 3] exactly. **(b)** Evaluate the Berry-Esseen bound for n = 50. **(c)** How large must n be for the bound to guarantee the CLT error is below 0.01?概率困难derivation未尝试免费469Geometric Mean of Random Multipliers via LLN and CLTAn investment grows by a random factor each year: in year i, the portfolio is multiplied by X i, where X i are i.i.d.\ with P(X i = 2) = P(X i = 4) = 1/2. After n years, the annualized growth factor is the geometric mean G n = (\prod i=1 n X i ) 1/n . **(a)** Find \lim n G n almost surely. **(b)** For n = 100, use the CLT to approximate P(G 100 > 3). You may use \ln 2 \approx 0.6931, \ln 3 \approx 1.0986, and \Phi(1.70) \approx 0.9554.概率困难derivation未尝试免费470Asymptotic Distribution of the Sample MedianLet X 1, \ldots, X n be i.i.d.\ Uniform (0,1) and let M n denote the sample median (the middle order statistic for odd n, or the average of the two middle values for even n). The asymptotic theory of order statistics gives n \,(M n - m) \xrightarrow d N\! (0, 1 4[f(m)] 2 ), where m is the population median and f is the density at m. **(a)** For Uniform (0,1), identify m and f(m), and state the asymptotic variance of n \,M n. **(b)** For n = 400, approximate P(M 400 > 0.54). You may use \Phi(1.60) \approx 0.9452.概率困难derivation未尝试免费475CLT with Estimated Variance via Slutsky's TheoremLet X 1, \ldots, X n be i.i.d.\ with mean and finite variance 2 > 0. Define the sample variance S n 2 = 1 n-1 \sum i=1 n (X i - X n) 2 and the studentized statistic T n = n \,( X n - ) S n . **(a)** Using the LLN and Slutsky's theorem, show that T n \xrightarrow d N(0,1). **(b)** In a study with n = 100 observations, you find X 100 = 12.5 and S 100 = 3.0. Assuming the true mean is \mu 0 = 12, approximate P( X 100 > 12.5) using T n. You may use \Phi(1.67) \approx 0.9525.概率困难derivation未尝试免费893Two-Segment Demand Total 3A forecast splits annual streaming hours into Segment A with 420000 entities at 1.8 events each and Segment B with 90000 entities at 4.2 events each. What is the combined annual estimate?脑筋急转弯中等数值题未尝试免费894Two-Segment Demand Total 4A forecast splits annual support chats into Segment A with 15000 entities at 2.5 events each and Segment B with 3000 entities at 6 events each. What is the combined annual estimate?脑筋急转弯困难数值题未尝试面试订阅895Two-Segment Demand Total 5A forecast splits annual lab samples into Segment A with 800 entities at 13 events each and Segment B with 1200 entities at 5 events each. What is the combined annual estimate?脑筋急转弯困难数值题未尝试面试订阅900Implied Unit Count 5Annual charging sessions are modeled as units * 330 events per unit per active day * 96 active days. If the annual total is 2851200, how many units are implied?脑筋急转弯困难数值题未尝试免费1385Two-Leg Basis-Point Net 5A book has two PnL legs on the same notional 15000000: one contributes 5 bp and the other contributes 8 bp. What is the net dollar PnL?脑筋急转弯困难数值题未尝试免费1402Exchange Fee Pool 2A desk proxy for daily transaction fee revenue is shares traded * fee per share. If the total is 3.22 million and shares traded is 9200000000, what fee per share is implied?脑筋急转弯简单数值题未尝试免费1403Swap Ticket Notional 3A desk proxy for total swap notional processed in a day is trade count * average notional per trade. If those inputs are 145000 and 68 million, what estimate results?脑筋急转弯中等数值题未尝试免费