Joint density

[zulkfal_1981]

Am a software engineer, and not know much about statsitics but have to explain my younger brother, please help me with this, thanks

X1 and X2 have the joint Pdf give as

f(x1,x2) = 1 0<x1<1 ; 0<x2<1
0 elsewhere

what should be the joint density of Y1 = X1+ X2 and Y2 = X2
and the marginal density of Y1.

[zulkfal_1981]

plz help me on this, i tried but he said that it has to do some thing to find some range plz.

[Etienne]

I think that you just have to use the Jacobian formula :

f(y1+y2,y2)=f(x1,x2)*1 because the Jacobian of the transformation is:

1 -1
0 1

and the determinant is equal to one. So the joint density of Y1 and Y2 is
1{y1-y2 in [0,1]}*1{y1 in [0,1]} where 1{.} is the indicator function.

To obtain the marginal density of Y1 you just need to take the integral of the joint density over the values of y2. You will probably need to be carefull with some min max things.

Etienne