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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2863A Batch-Size Compound Poisson Desk FlowTrades arrive according to N\sim Poisson ( ). Each trade contributes a batch size B taking values 0,1,2 with probabilities 1/2,1/3,1/6, independently across trades and from N. Let \[ S=\sum k=1 N B k. \] Find the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2864Exponential Random Intensity Gives Geometric CountsA latent intensity \Lambda is Exponential ( ) with rate . Conditional on \Lambda, the count N is Poisson (\Lambda). Use MGFs to identify the unconditional law of N, and compute E[N].概率中等derivation未尝试面试订阅2865A Stable Law with No MGFSuppose X 1,X 2,\dots are i.i.d. with characteristic function \[ \phi X(u)=\exp(-c|u| 3/2 ),\qquad c>0. \] Show that \[ n -2/3 (X 1+\cdots+X n) \] has the same distribution as X 1.概率困难derivation未尝试面试订阅