# Distribution of 2 Random Variables

• November 29th 2009, 03:11 PM
statrel
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.
• November 29th 2009, 09:51 PM
matheagle
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.