Question

Consider the bivariate function f(x.y) = (x + y)/3 for 0< x< 1 and 0<y< 2 and f(x.y) = 0 3. otherwise. (a) Show that f(xy) is a density function. (b) Find the probability that both X and Y are less than one. (c) Find the marginal densities of X and Y and show that they are not independent. (d) Find the conditional density of X given Y when Y = 0.5.

Answer 1

Page f(x, y) = xty for osxel 3 0 yea flasy) = 0; otherwise - f(n) is non negative also 546 xyz Se+(4+2) 1 1- I Hence it is a probability drsity function

o <x< 1, OLYCI Jo Jo 3 Sy fable,y) a fil xty dedy STO (+24]. $94+) 9Marginal drastyef * {X2+ xtydy 4 L - SHOT ON MI A1 MI DUAL CAMERA ata if ocuci o otherwise

Merginal deity of Y = fyly) - (x plony) abe - 2y +1 o Olya otherwise Both marginal density of X ad y contain value of X and Y respectively. So they are defendent" on values of Xand 4 They are not independent I Plugy - nt y fl 79=0.5 fyly 30 2 SHOT ON MI A1

Page No. But y = 1 2[ 37 - =) anti Conditional 2 dersity