第 1 / 1 页
非代码面试题
显示 11 / 11 道匹配题目
答题状态：未尝试未正确已正确
ID题目领域难度题型进度权限
215Distribution of Dice Sums via Probability Generating FunctionsLet X 1, X 2, \ldots, X n be iid rolls of a fair d-sided die, so each X i is uniform on \ 1, 2, \ldots, d\ . Let S n = X 1 + X 2 + \cdots + X n. (a) Derive the PGF G X 1 (s) = E[s X 1 ] in closed form. (b) Write the PGF of S n and use it to derive E[S n] and Var (S n). (c) For n = 3 fair six-sided dice (d = 6), use the PGF to find P(S 3 = 10). (d) Explain how the coefficient-extraction approach relates to the classical stars-and-bars counting with inclusion-exclusion for this problem.概率困难derivation未尝试免费2816PGF of a Binomial VariableLet X\sim Binomial (n,p). Derive its probability generating function G X(s) and use it to recover E[X].概率中等derivation未尝试面试订阅2817Sum of Independent Poisson CountsLet X\sim Poisson (\lambda 1) and Y\sim Poisson (\lambda 2) be independent. Use PGFs to identify the distribution of X+Y.概率中等derivation未尝试面试订阅2818Poisson ThinningSuppose N\sim Poisson ( ) and each event is independently kept with probability p. Let K be the number kept. Use PGFs to identify the law of K.概率中等derivation未尝试面试订阅2825Compound Poisson With Geometric Batch SizeLet N\sim Poisson (2), and conditional on N, let \[ S=\sum i=1 N B i, \] where the B i are i.i.d. geometric-on-\ 1,2,\dots\ with parameter 1/2, so P(B i=k)=2 -k . Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅2829A Geometric Number of Bernoulli TrialsLet N have the geometric law on \ 0,1,2,\dots\ with P(N=n)=p(1-p) n. Conditional on N, let \[ S=\sum i=1 N X i, \] where the X i are i.i.d. Bernoulli(q). Find the PGF of S and identify its distribution.概率中等derivation未尝试面试订阅2832Binomial Number of Trade BatchesLet N\sim Binomial (5,0.4). Conditional on N, let \[ S=\sum i=1 N B i, \] where each batch size B i has PGF H(s)=0.5+0.3s+0.2s 2. Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅2833Zero Total in a Compound Poisson Batch ModelLet N\sim Poisson (3) and let the i.i.d. batch sizes B i satisfy \[ P(B i=0)=0.2,\quad P(B i=1)=0.5,\quad P(B i=2)=0.3. \] If S=\sum i=1 N B i, compute P(S=0) using the PGF.概率中等derivation未尝试面试订阅2835A Geometric Parent Count With Even-Sized BatchesLet N have the geometric law on \ 0,1,2,\dots\ with P(N=n)=\frac13 (\frac23 ) n. Conditional on N, let \[ S=\sum i=1 N B i, \] where P(B i=0)=P(B i=2)=1/2. Compute P(S=4) using the PGF.概率中等derivation未尝试面试订阅2838Sum of Geometric CountsLet X 1,\dots,X r be i.i.d. geometric random variables on \ 0,1,2,\dots\ with \[ P(X i=k)=p(1-p) k. \] Use PGFs to identify the distribution of S=X 1+\cdots+X r and compute E[S].概率中等derivation未尝试面试订阅2840Poisson Number of Geometric Ticket GeneratorsLet N\sim Poisson (3). Conditional on N, let \[ S=\sum i=1 N B i, \] where each B i is geometric on \ 0,1,2,\dots\ with success parameter p=1/4, so P(B i=k)=\frac14(\frac34) k. Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅