Question

f(x,y)=0 2. (20 marks) Suppose X and Y are jointly continuous random variables with probability density function fc, 0<x<1, 0<y<1, x + y>1 else a) (2.5 marks) Find the constant, c, so that this is valid joint density function. b) (5 marks) Find P(Y > 2X). c) (5 marks) Find P(X>0.5 Y = 0.75). d) (5 marks) Find P(X>0.5 Y <0.75). e) (2.5 marks) Are X and Y independent? Justify your answer citing an appropriate theorem.

Answer 1

fca, y) = 5c 0<x<1, ocy<1, 1+471 a) sfony) dy dx , » Hic dy dx = faly.doc = c [-(-20) dx = 1 -) C[ 321-C1") = 1 =) c=2 o bl-x - f(x,y) = 12 ochel, olyki, arty> Marginal pdf of . fx(x) = f(a.s)dy = 12 dy = 2 [y ]... 22 - 2 fx/n) =) 2* 02012! To ;ow. Marginal pelf of YT fyly = f(ns) se - Jody - 2 (a)iny = 24 fyly)= { 24 orgel . - I-y e) conclitional pdf 0.75 de f (xl y = y) = f(ny) . 2 - - - c) PIX>0:5|2015) f(x|9:0:15) dx 16.05) 03:56 P[x>0.5|Y20-25) = 0.6667 0.5 o.s

f(x,y) & +(2).fly ...X and Y are not independent. b) P[x72x] . 2x 4/2 14/2 Ply>2x] = fcay dady = 12 dady o o = 12 (1) dy = Jydy = 1/2 DO O IP (Y>2x] = .0.5 d) Plx>o slY20.25) 0.75 0.75 y PLY.20.75] = 575 for lyddy = Jaydy = [y?) [x70s, by 20:25] = 1 [ ] f(my) dx]dy = J J 2 dndy og reddy 2 (4-0-5]dy 242. asyo75 0.5 0.5 6.5 os 0.5 Plx>0.5,4 2075] = 0.0625 Plxzos ly 2015 0-0625 0.5625 0.1111