Question

3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.

Answer 1

TOPIC:Conditional distribution,expectation and marginal pdf.

Given that, X~ Exp (A=8 and, [[y/x] Exp (x-x) So, the conditional pdf of (Y1X= 2) is (C11X =») « Exp (3-2) d (7)= { xe ylx=x ; Jon, J7O. o ; otherwise. - where xro. t conditional pol of (x1x²x). (Ana ③ we need, => [Ylx=x] Ju do dy x=x (7) dy. 2 Sage dy. 2-1 dy 2. (2) 2 2 K-1 -py e dy There, ka2 p=x. [(K) pk There is = 1 {: P(2) -(2-D!-!

So, E(Y/X=) == which is troue, for all x 70. Hence, E (Y(x) = 1 1. And o Again, the joint pof of (x, y) is Ylx=x where tx(x) = marginal pdf of x. =) de (x) = { se ; nyo. ; otherwise. Thus, we get * txiy (x, y) = { xe -sx де. ; 270,470 0 otherwise. = xy(x) = { xs. e us. 2**(4+4) ; 20, 470 ; otherwise,

→ Thus, the marginal pdf of sy is → > 67(4) = S by (3, 3)dx 2 -x(s) dx. Size Pata de -8. dx. xe xestal 00 ky - px, e (2) r(K) ok - Here, k=2, Þ=(+3) i formy 7 you (Sta2 Hence, ty (7) = { $ (8+)² i 70 oney jotherwise, - Where to sro.