Question

The joint probability density function of X and Y is given by fxy(x,y) = {6xy 0 5x S1, o sy s va 0 otherwise

(a) Sketch the region in the (x,y) plane where ??,?(?, ?) ≠ 0.

(b) Find the marginal probability density functions ??(?) and ??(?) of ? and ? respectively.

(c) Are X and Y independent?

(d) Find P(Y>X).

(e) Let y be some real number in the range 0 ≤ y ≤ 1. Find the conditional probability density function ??|?(?|?).

(f) Find ?[?|? = ?] (where ? is some real number in the range 0 ≤ ? ≤ 1).

Answer 1

Х Solution: The joins paf of смо у сvси А): S 6x40<x< 1 osyssa Otherwise ЧЕ for - 0 (1,0) Х

The marginal pdf of a as (x,y) dy nux FB : Si ху Je = 6x Syoly (427 we = 6x = 372 f(x 2 = {322 Lo Osx=1 Olturwise

The margiuae pof of you fyll) = f (xy dx 92 =by Sada E by by ETA = 34 (1-44) f() = {34 (1-44) 1 OLY=1 Ć Otherwise O

@ the joins pay of x and you Array = 6xy We have a 2 We Acerty= 332 3.468-444) From sepuolit, duo digunhiên gues To buyFoto Hencer & and are not independars 3

nux = Foxy) oly dx @pCY21) SS 6x su dy de ona a dx X =3 3 x (x-x2) dx =3602 (1-x x) dx =3 Bela (3,2) = 3 13 12 382 151 хаха Heuca P(%>x) = 4