Question

Let X be uniform on [0, 1], and let Y be exponential with rate λ, so that P(Y ≥ t) = e ^(−λt ) t ≥ 0 and 1 if t < 0

Assume that X and Y are independent, and define W = X + 2Y .

a) For any w ≥ 0 and x ∈ [0, 1], compute P(W ≥ w|X = x)

b) By undoing the conditioning on X, use the result from part (a) to compute the cdf of W.

Answer 1

x~ Unifolm (10, 13) yn Exponential (at) @ P(W zw/x=x) = P(x+2yZw/x= x) = P(x+2yzw) (this is valid because =P(2yZW-x) = P(yz W-x) X and Y are un dependent) (Since Plyzt) = ent to = len (W-X)/2 Wax 20 W-xco. Wzx P (W=w/x=x)=, e (W-X)/2, wzx W<x. I

(6) We can write P(wz w) = P(wzw/x=x) fy() dx. where fy(x) is the probability density function of x <x > P(WZW) = P(wzw/x=x) (1) dx if oewel 2 + x 1x/2 W + = è dw/2 I-W etwiz letala dx + x = zdarz wys + - - edulaz (entula,) +1-w = 3 - 50W/2+1-W P (W zw) = 2 (1-et W/2) +1-w if os W=1

P(W=w)= | P(w z w/ x=x) dx. - (W-x)/2 dx if w> = edula l'etele dx. - edula ex12 | - edula e diz) lo = endula ac etia_i). = 2 ( en (W-1)/2 ēdw/2) if wsi Therefore P(w=w) = ( 3 (1 - 0 211-etul2)+1- w if Osw=1 2 ( 81 (W-1/2 etwa) if war CDF of w P(w sw)=. P( Wsw)= |- P(wzw) W-2 (1-ed W12) if oswal 11-2(ew-1)/2 e dulz) if wal