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~
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~
the constraint is $\displaystyle x+y\le 1$
the two rvs are uniformly distributed over that triangle in the first quadrant
so $\displaystyle f_X(x)=\int_0^{1-x}2dy=2(1-x)$ on 0<x<1
$\displaystyle 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
$\displaystyle 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.
$\displaystyle P(0<Y<3/4|X=0.5) =\int_0^{.75}f(y|x=.5)dy = \int_0^{.5}2dy=1$