第 1 / 2 页
非代码面试题
显示 20 / 35 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
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224Compound Poisson Distribution: MGF and MomentsLet N \sim Poisson ( ) and let X 1, X 2, \ldots be iid discrete random variables (independent of N) with PMF P(X i = j) = p j for j = 1, 2, \ldots and MGF M X(t) = E[e tX 1 ]. Define the **compound Poisson** sum S = \sum i=1 N X i (with S = 0 when N = 0). (a) Derive the MGF of S. Show that M S(t) = \exp\!\big( (M X(t) - 1)\big). (b) Use the MGF to derive E[S] and Var (S). Express your answers in terms of , E[X 1], and Var (X 1). (c) Alternatively, derive E[S] and Var (S) using the tower property (law of total expectation) and the Eve's law (law of total variance), conditioning on N. (d) **Application:** An insurance company receives claims at a Poisson rate of = 10 per day. Each claim size is \1000 with probability 0.6 or \5000 with probability 0.4, independently. Find E[S] and Var (S) for the total daily claims S, and compute the standard deviation.概率困难derivation未尝试免费233Mean and Variance of the Lognormal DistributionLet X be a lognormal random variable, meaning \ln X \sim N( , 2). Using the moment generating function of the normal distribution, derive E[X] and Var (X).概率中等derivation未尝试免费235Chi-Squared Distribution from First PrinciplesLet Z 1, \ldots, Z n be i.i.d. N(0,1) random variables and define Q = \sum i=1 n Z i 2. (a) Derive the PDF of Z 1 2 using the change-of-variables technique for transformations of continuous random variables. (b) Show that Z 1 2 \sim Gamma ( 1 2 , 1 2 ) by matching the PDF from (a) to the Gamma family. (c) Using the fact that the sum of independent Gamma random variables with the same rate parameter is Gamma, state the distribution of Q. (d) Derive E[Q] and Var (Q).概率困难derivation未尝试免费1520Local Hedge Multiplier 5Use a second-order Taylor approximation around 0 to estimate (1+-2x) (1/2) * (1+5x) (1/2) at x=1/100.数学困难derivation未尝试面试订阅2027Log Barrier Jensen Direction 7A utilization score explodes as it nears 1, so the convexity direction matters for stress design. Let u(x)=-ln(1-x) on x<1. If U is random and almost surely below 1, compare E[u(U)] and u(E[U]).数学中等derivation未尝试免费2035Funding Buffer Gap With Unequal Scenario Weights 15The high-leverage state is rarer, but still materially affects the convex average. A funding-buffer model uses phi(L)=1/(1+L). Suppose L takes values 1 and 4 with probabilities 1/4 and 3/4. Compute E[phi(L)] and phi(E[L]).数学困难数值题未尝试面试订阅2043Log Carry Gap From Two Scenarios 23The desk wants to see the exact concave Jensen gap, not just the inequality direction. A desk scores carry through psi(x)=ln(1+x). Suppose X takes values 0 and 3 with probabilities 1/2 and 1/2. Compute E[psi(X)] and psi(E[X]).数学中等数值题未尝试面试订阅2841Signed Compound Poisson Order FlowMarket buys arrive as +1 and sells as -1. The number of trades in a minute is N\sim Poisson ( ), and conditional on a trade, the sign is +1 with probability p and -1 with probability 1-p, independently. Let S be the net signed flow in that minute. Find the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2842Compound Poisson with Exponential SeveritiesClaims arrive according to N\sim Poisson ( ). Claim sizes X 1,X 2,\dots are i.i.d. Exponential ( ) with rate , independent of N. Let S=\sum i=1 N X i. Derive the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2843Gamma-Poisson Mixing Produces Negative Binomial CountsA latent intensity \Lambda is Gamma ( , ) with shape and rate . Conditional on \Lambda, the count N is Poisson (\Lambda). Use MGFs to identify the unconditional distribution of N.概率困难derivation未尝试面试订阅2844A Geometric Number of Exponential StagesA task takes a geometric number of stages: N has support \ 1,2,\dots\ with P(N=n)=p(1-p) n-1 . Each stage duration is i.i.d. Exponential ( ), independent of N. Let T=\sum i=1 N X i. Use MGFs to identify the law of T.概率困难derivation未尝试面试订阅2845Recognizing a Shifted Poisson from Its MGFA random variable has MGF \[ M X(t)=\exp\!\bigl(2t+3(e t-1)\bigr). \] Identify the law of X, and compute E[X] and Var (X).概率中等derivation未尝试面试订阅2848Reading Covariance from a Joint MGFSuppose \[ M X,Y (s,t)=\exp\!\bigl(2s-t+2s 2+3st+\tfrac52 t 2\bigr). \] Compute E[X], E[Y], Var (X), Var (Y), and Cov (X,Y).概率中等derivation未尝试面试订阅2849Difference of Two ExponentialsLet X,Y\overset i.i.d. \sim Exponential ( ). Use MGFs to identify the law of D=X-Y.概率中等derivation未尝试面试订阅2856Compound Poisson with Gaussian JumpsLet N\sim Poisson ( ) and let Y 1,Y 2,\dots be i.i.d. N( , 2), independent of N. For \[ S=\sum k=1 N Y k, \] derive the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2858Reading a Laplace Law from Its Characteristic FunctionSuppose a centered random variable has characteristic function \[ \phi X(u)= 1 1+b 2u 2 . \] Identify the law of X, and determine its MGF on the domain where it exists.概率中等derivation未尝试面试订阅2859MGF of the Sample Mean of ExponentialsLet X 1,\dots,X n be i.i.d. Exponential ( ) with rate , and let \[ X n= 1 n \sum i=1 n X i. \] Find the MGF of X n, and recover E[ X n] and Var ( X n) from it.概率中等derivation未尝试面试订阅2862Factorized Joint MGF Means IndependenceSuppose \[ M X,Y (s,t)=\exp\! (s+2t+ s 2 2 +2t 2 ). \] Identify the marginal laws of X and Y, and determine whether they are independent.概率中等derivation未尝试面试订阅