Question

2. (30 pts) Let X and Y be independent rv's with pmf Pois(41) 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 = x ~ Bin(x,p). Find the distribution of Y.

Answer 1

Given that X and y are independent random variables coith pmf poisson (10) & pofsson (12) resp. xn poi scon (do) , you polsson (19) p(x)= elit . ply)= etage X! Y! @ we have to find the distribution of z =xty. 2=&ty ; we have for z>o. P(x+y=2) = 8 plx +y = 2, x=2) PC4-7-, 4-2) PCY=21, 2+) ply=z-2). P(x=) (xay are Independent) - Nuling will E 2-1)! ! (-i)! elu tie) a birtda) .

we know that & ncept qnok = (2+pja - Chitha) (x,xxx} 20x+y follocas poisson Cort de). we have to find the distribution of xlx+y Xop(X1), yop(x2) xty N P(x + Xx) (from @) I let X=r; x+y= P(x=rl x+y= n) - pcx=r n x+y=n) X+ Vé ) a PCX=r ny=nr) p(x+y=0) = P(X=r). PCY=n=) P(x+y=n) X and y are independent Since - r . (M-1)! Ehith atten

=é (81486). * AC Nut? ! T!60)! (AIGAS)Cartas)" = n! v M Cht) h r! (n-v)! (Artdo) r nr 2 here you see have to co me and then = (s) proq may where p. A la moda dit de is a Brnomial distribution with parameter ne pa nad (Xlx+y) N Ben (n, p).

© we are given that, p(x) = el. ! yly w Bin (%,P). ply=r/X34) = ºv) pF (1-)**; osper 530,12,... 8. p(x= x nyar)z plv). PC Y=rx=4) =ēm,*. Or pr. co-pjects. ply=r) = The unconditronal desti kaukion ofy Carv) p"Co stan] 56"[14, 6)** (veter] A , P x=rl xl X! =&M[Esenyum pacote] = 5" (609"71, Copen

* EMAIO 26=8 a ēdi (diplo 1,(1-P) = 291 + X– dyp : Clip) Eē die CP) ? -8 0 ,1,2,... a pomif of poisson distribution .. Hence y is a poisson variate with parameter Nip.