Question

Problem 3: X and Y are jointly continuous with joint pdf 0<x<2, 0<y<x+1 f(x,y) = 17 0, Elsewhere a) Find P(X < 1, Y < 2). b) Find marginal pdf's of X. c) f(x|y=1). d) Find E(XY). dulrahim

Answer 1

Marginal pdf of a random variable can be found by integrating the joint pdf with respect to the other variable

Joint pdf of row. x and yeis given as frays - S 3 xy, 04842, OLYL xxl lo , ether wise (a) P [X21, 122] = $$$? To diy dz = $ (2) dx - 1 372 (120) dx = À (6) Marginal of gl* = { fenys dy = texiysay

gixs = pt 2 ny dy = [texy?-o] gex) = 3x (x+1) 27 o(XC2 34 (e) fexly=1) marginal of Y heys = I fear y dx = g sxy dx - (2) 3414-01 бар, f(xly=1) = f(x,y=1) f(y=1) fla. y=1) hey=1) frx\y=1) = 2

(fex 13-1) = / (d) ECTY) ECHN) = f l xy femeys dady - se pot xy. 3 xy dx dy => * 4 'de J x2 ( x3+1+3x2+3x) dx = 1 1 4 8 1x5 + x² + 3x² + 3x3] else CZ + +385+ 3x] to C2 + g + 6 + 45 = A ( 54 0 st60 + 1152772 0 ] Ecxy) = 455