# Thread: Distribution of 2 Random Variables

1. ## Distribution of 2 Random Variables

Hi,

I have a joint distribution function

f(x,y)=12xy(1-y), 0<x<1, 0<y<1

and need to determine the distribution of Z=X+2Y.

I know that I can do this by finding F(Z) over different regions (i.e. 0<z<1, 1<z<2, 2<z<3) and then differentiating for f(z), but am having some trouble figuring out what my different regions of integration should be. Any help would be greatly appreciated.

2. You can do this either via F or jacobians.
X and Y are independent, but that isn't of any help here.
If you want to go cdf route then....

$F_Z(a)=P(Z\le a)=P(X+2Y\le a)$

and you now need to draw the line x+2y=a and see how it slices through the unit square.
The region of interest will be below that line, in other words,
between the origin and that line.