You are given a joint pdf of two random variables where f xy

You are given a joint pdf of two random variables, where
f x,y) = 1/4(y + 2x) when 0<x<1 ; 0<y<2
f(x,y) = 0 otherwise

a) Prove that this is a valid joint pdf.  
b) Calculate the marginal density of Y.  
c) Use the marginal density of Y to find P(0<Y<1).
d) Calculate the marginal density of X.
e) Calculate the conditional density of X given Y.

Solution

a) integral 1/4(y+2x)dxdy

= integral 1/4 * ( yx + x^2)dy   Putting 0<x<1

= integral 1/4 * (y+1)dy

= 1/4 ( y^2/2 + y )

=1/4 (2+2)

= 1

Hence Valid PDF

b)

fY(y) = integral 1/4(y+2x)dx

= 1/4 ( xy + x^2)

c)

1/4 ( xy + x^2) 0<Y<1

= 1/4 ( x + x^2 - x^2)

= x/4 Answer

d)

fX(x) = integral 1/4(y+2x)dy

= 1/4 ( y^2/2 + 2xy)

e)

P(X/Y) = 1/4(y + 2x) / 1/4 ( xy + x^2)

= (y+2x)/(xy +x^2)