Question

Let X and Y be independent rv's with pmf Pois(11) and Pois(12), respectively. (a) Find the distribution of Z = X+Y. (b) Find the distribution of X X +Y. (c) If X Pois(11) and Y|X = r ~ Bin(x,p). Find the distribution of Y.

Answer 1

To delepmine the distpibrtion of 7-X+Y,we can use probability generating funetion(as patssen distribuion is disepete) Fon indepent X and Y pandom vapiable fello oing Pai (A1) and Poi (2) respectively PGF(pobabirty aenenating function) of x is Px- Elt]-t PIx] X0 =e tohich Ts an exponen tial gepies = e- 1 eta1 = e- C1-t Sirmilael Py-E[t] = e-A2(1 -+). Noto, let us think about PGF af Ze x+Y Pz()Et1 -E[t E[ mdependent Tx and Y ape - E [t 1. E [t1 =Py(t). Py(t) = e- Aa (1-) e -A2 (1-4) = e-(a1+a2 (1 -4) This seems lo be a PGiF of poisson (a1+a2) disteibuln Since PGF uniquely identifies. distoibutron , thus Z (x+y)Pot (1+a2)

(b) Hepe ,oe need to ind the distptbution of X , e P[xIxtyl Now, P[x= |x+y=2 PTx-x, x+y=7] PE%-, y-- P[x+y=7] P[x-].PIy- -x] x and y ane independent PExty 7] (s.- 7! x1 (- -Aa+ 22 a1+a2) (7-) 21(2-x)! 2-X+x 7-X (:) A2 (21+22 21+A2 R Binamial . [xIxty] 21+22

(e) X Pois (1) YIxx Bin. (x,p) We want to ftnd the distpibution of Y,e PLy] PLY-y-PIx- x, Y-y]elx-x,y- o the Joint Pnobability mass function of Xand Y X:0 PEYIX.PX-] PCY=gIx-x] PLY-Y.X PEx 1 .e- -Σ (x-y)! (1-) g- Σ e-ad (x- = P.e (1.- -y (- x-0 -21 (1-H)a) exponen lial sepfes sum .e-a1 eC1-)21 . - 21 (1 -1+p e ! (pa e A1b :. PLY 1- e-21 (ab). YPois (21P)