Question

0 2.2.8. Suppose X1 and X2 have the joint pdf I e-le-22 21 > 0, X2 > 0 f(x1, x2) = elsewhere. For constants w1 > 0 and W2 > 0, let W = W1X1 + W2X2. (a) Show that the pdf of W is 1 -(e-w/wi – e-W/W2) W > 0 fw(W) = { W1-W2 C 10 elsewhere. (b) Verify that fw(w) > 0 for w > 0. (c) Note that the pdf fw(w) has an indeterminate form when w1 = W2. Rewrite fw(w) using h defined as W1 - W2 = h. Then use l'Hôpital's rule to show that when w1 = W2, the pdf is given by fw(w) = (w/wị) exp{-w/wi} for w > 0 and zero elsewhere.

Answer 1

Solutian: Gaven that and 12 Joint pdt a, and 370 21 efininanew Yandom Variable, Now nmpute tc p of W Let y WsX2 om one -to-oe bansfomation gaaledly and = X W-y w2 Jacabian tansfoalian Sw J= we W y J = and y i5, Then, e jon Pat a w w-y a wy(wy)- w2 w-y

Naw, ampute the warginal pof f Sspoatnel bolow, W is P(w)=fu(w) w-y op W2 w-y y wiwz X X Wiw2 W70 Wi-W2 (b df w s, Gaden T fu (w) w2 then w Lo and Exponential unctian Sxp wo

(C) Cdan, then pdf af w beluwes Byusig hHpitalule, f(w WI-h 230 Itm 1-70 F0-lim 7-50 h By using ptal ule f(w) Thank You