Question

2. Suppose X and Y have the joint pdf fxy(x, y) = e-(x+y), 0 < x < 00, 0 < y < 0o, zero elsewhere. (a) Find the pdf of Z = X+Y. (b) Find the moment generating function of Z.

Answer 1

pase 4 Question suppox Xby hare the joint pidid Sanay) Cara ay Find the podt of z=xty. OLX La & OcYLO . Find the moment bonenting function of z. Sot": At first we will solution of auestion (b). @ Moment genenating function of 2= x+4:- Given that, the joint density of xfy is for aty, our La OLYLD The marginal density of o is I OLKLO tory La fx cu 2 f joint density af (ww.dy ) - ويله رومه 엘s y z něk [eg]." = new by z ? sex [ 0-1 ] Sine sēr you to ed=0) manginal density of x is given by fx (ry a ex = 1 . elix, olxoo lo, othnaise

Page 2 Tror forlanginal density at y is given by Fra Joint density at xbx.de po Suray.de seeney, de ode .hu arey [22] 2 penkt: életěty Hence, the manginal density of y is gian by fy (9) 2 (1 e 2 y ocyer 0 Otherwise --- to row, we know if p exprential (d) with nate d. Then the pidid of p is given by forkp) - této oszcato Also we know X & Y are independent pandom caniasse if joint distribution of xd y = production of their marginal - ces for say = fracas. foly) Here, fear > ē (x+y) Result: fx cm. Lyle) e manginalety il joint density of xby - marginel of x & Hence, by Result of we get l y ame independat Randon variables

paye! } . Hene, from ly using we get. Xv exp (A) & you exp (1). Also x&e or independer il x & y are independent and fullous exponential (A) distribution dow, suppose it por exponential (1), then the moment generating function of p is given by - Mp (ta Eleto) = Seta. padi Lodpldp ine, the pididate = fope Aldo [illow dep, pro adje --t)P dp 2 - 1 4 [ e (6) p] - City té el - 10-2] = 1/2 » Mp (t)e mate van (6)where t depends on convergence condition, il Here, an exp (1) & you exp (1) . . taking =1 to get the moment genenebing function X&Y in Malta Myct) = (+) ---- Let, Z=xty. Then then moment Gespenating function of z is given by M₂ (t) = E(et) I 21 in ©

paye: 4 Hene, X & Y are independent at the are independe So, Eletro ett) Eletro). El et4) So, M₂ (t) = E(et). Eletuj - Mx (t). Mylt) since, from t we get = - 1 [me (1) Mult) t-t] th Hence, the mgf of z is Mz (t) = (tt) Ans) swhere 4121, @ Find the probability density function of a? we know, if for Gonna (218) . Then the pidid l moment generting function (ingt) at are given sy. - convengone condition, al roku 1941 Behu 210 / (3) & Multja a to Here, mig-f of z ism So, using o we Get zw Coma (1, 2) distribution. and pedes at z is trium by l in take, d=1&p=2) 2 -12 2-1 e 2 02220 0 Co otherwise - lze z , id o LaLD o, Othenuise ( since Cr(n) = (n-1)! Ans)