1. ## simple questions need answer # 3

3. Let the joint density of (X,Y) be given by f(x,y)=2, 0<X<1, 0<Y<1-X
a) Calculate marginal density f(x),
b) Calculate P(0<Y<3/4|X=0.5).

Thanks~

2. the constraint is $x+y\le 1$

the two rvs are uniformly distributed over that triangle in the first quadrant

so $f_X(x)=\int_0^{1-x}2dy=2(1-x)$ on 0<x<1

$f(y|x)= {2\over 2(1-x)}={1\over 1-x}$ on 0<x+y<1

THIS IS UNIFORMLY DISTRIBUTED on any line for a fixed x

$f(y|x=.5)= 2$ on 0<y<.5

so P(0<Y<3/4|X=0.5) is a silly question, the answer is 1.

$P(0