Question

2. Let the random variables X and Y have the joint PDF given below:

Let the random variables X and Y have the joint PDF given below: 2e -0 < y < 00 xY(,) otherwise 0 (a) Find P(XY < 2) (b) Find the marginal PDFs of X and Y. 1 (c) Find the conditional PDF of Y|X = x (d) Find P(Y<3|X = 1

(a) Find P(X + Y ≤ 2).

(b) Find the marginal PDFs of X and Y.

(c) Find the conditional PDF of Y |X = x.

(d) Find P(Y < 3|X = 1).

Answer 1

Solution Given that -f(,y) = s2e1 roeg osiey Zoo otherwise a) P(x+y< 2) 2 2-2 P(x+y=2) = P(452-x) = 2e e dy.da Bercy and 22:2439) resz")an = 2(32-24) =zf78e1to+ +] .. l-ě4-4ė? : 0.4403 P(x+y = 2) = 0.4403

X the mouginal w is The çrojoc e ydy: (2014)" = zc22 . The marginal pdf of y is fly) jeezy dx = [-zeazy)= 200. rey 2e conditional p.dif Of Yx = x fylx (/x) = f(x,y) 2e2-y B fx(x) 2e2x sy ex d) pCY231*= 1) Putting xal in above conditional pdf gives fylx(41) = f(x,y). ..cy - fully Ply<3\x= 1) - Serialy In) ay Sečudy. = efeys = ele'-ě3 = 0.864-66. — Thank you =