## Question

###### Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is...

Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1

## Answers

Lucky Page No: Date: 1 - Xu X -- X4 gie independent Random variables ne Lexi_ 70 let Let XuXQ1 -. X are poisson Tandem variables with means_5e114 42 Letxo Poisson (20: e mgf of x is. Joacta Ecetx) - Dete ēd g? 21=0 2 - 230_ ! ettet)Lucky Page No Date : 1 'xit X2+ Xg+X4 32. Ixi Mgf of y is My Lt) - Mxit to the two . .. .Ele+ (x1 + x2 + x3 + x4)) - Efetxi). Ecetx2) EletX3) Eletry --50-et) -(-ett -4 (1-ett 32(t-1) नी-et) (छकाया) 10 - ele (1-et) → Ya Poisson (12) ECY)= V(Y) = 12.Lucky Page No: Date: xv. Geometric (D). Ilona a Myct)Eletx) - I ate (l-p)?.p. = p. (-)et)". = p [1-(1-P)et] 1-11-Det I New Y = Ixi Xi N Geometric cp-310). ITITI My (t) = Mxi(t) Mxg(t). Mxz (t) Men(t). SIP_98 [1-4-plet] |_31419 Ulucky Page No Date : which is mgf of negative bioomial dista Quittii parameters (4,314) III EL K (1-P) = 4x (114) _314 - 413 w(Y) = K (1-P) = 4x (14) 314.)

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