Question

2. Let the pair (X,Y) have joint PDF fxy(x, y) = c, with 2.2 + y2 <1. (a) Find c and the marginal PDFs of X and Y. (b) What are the means of X and Y ? No calculations are needed, only a brief expla- nation is required. (c) Find the conditional PDF of Y given X = x and deduce E|Y|X = x]. (d) Obtain E(XY) and compare it to E[X]E[Y). (e) Are X and Y independent? Explain. (f) Obtain var(Y) without resorting to integration. Hint: Use the fact that var(X) = var(Y) and (c).

Answer 1

· Lex gon)2 n 1-v2 of Sincexdy has Same distribution, Given that with 001) 000140,00 celndy = 1 (0:1) R endy C saint pof of (x,x) is fxx(1,4)= x ² + y ² LI la) we know that SS f(n,y) dndy =1 ŠS $cm,v) dedyr → SS condly Ssandy c = 4 (And) Margional density of x andy are given by fx (n) = 5 : f(ny) dy y=-11-22 = [TY** +JT-2022 R 2 area of R >> Cr=| > Co r 12 =r y=(1-12 ♡ F51-72 2 Si-n² -1LALI 2 JZ SI-42 no f(ny) en CJE fylw= s n=-S1-42 s .-1LYLI + [ 251-yz r T-yz +5142 distribution saime (6) clearly xpy are .: E(x) = In fx (m) din dn = / ENT- N1-22 -na dn - 20 s. Ely)=669) = 2/x0 Inson is old fun &6) 255) & g (ons-gin) q u an old fun so goncanzo for 8 (-1) - pon Since cs Scanned with CamScarin (x) 25(4) 20-

15 (C) Conditional. PDP of y given x on w fylx (Y/X=n) f(Y,X») f(x>n) 2 2 5 H 1-17 ---Ly UPAL COMPLAD RT 251-n² 1-n2 Now E [Ylxon] = Sufylt y fy 1x (7/xon) edy You si-n² 11-n2 251-1² -S-n2 [. ((1=n9=(1-2) =O 451-22² O (d) E [x] 2 Si-y N-n2 Sf ny f(ny) dn dy For nye a dnely La so no dx = 0 LAN n2-1-42 y=-1712 N=-51-42 47 -STA - 4²1 lo ECD) E[XY) 2.0 =B() BIY) e) X and Y are not independent Since fxy(my) & fix (m) & (9) Do not independent X, Y are Son (5) Now. E [72/x-n] s y² fo (Ylxan) dy yorim nanz 2. 2 m *} [us] ((1-x2) Si bias -x2) ST**+(1-2), o) -714 CS Scanned with CamScanner 13 (1-12)

Iron [ 5642E [E [x2) xon ZE [1-x² = s ('-E [Y:)) E[x] = kg (1-5 [x2]) => 3 E[x² 31-6[x] 2 6 [x²] = 14 - Var(x) = van(y) =E[x] -(EE»?) xdy have same listribution → E [x] 26[r] म 0 - - - (Any) CS Scanned with CamScanner