# Thread: change of variable help

1. ## change of variable help

1.
Let X~Geometric(1/4), and let Y have probability function:
pY(y)=
1/6 if y=2
1/12 if y=5
3/4 if y=9
0 otherwise

Let W=X+Y. Suppose X and Y are independent. Compute pW(w) for all w in R.

For this i am not sure i think its
summation from K=0 to infinity (PY=w-K)(1/4)(3/4)^K

where
(PY=w-K) = 1/6 if w-K=2
(PY=w-K) = 1/12 if w-K=5
(PY=w-K) = 3/4 if w-K=9
(PY=w-K) = 0 otherwise

Is this right?

2.
Suppose X has density fX(x)=(x^3)/4 for 0<x<2, otherwise fX(x)=0, and Y has density fY(y)=(5y^4)/32 for 0<y<2, otherwise fY(y)=0. Assume X and Y are independent, and let Z=X+Y.
Compute the density fZ(z) for Z.

For this is it
the integral from z-2 to 2 (x^3)/4 * (5(z-x)^4)/32 dx

Is this right?

2. The first one is a mess.
You can figure out the three cases of Y, but also I'm not sure how you want to write YOUR Geometric.
It can be written two different ways.

The second is best done by figuring out the CDF of Z.
There are two cases, we easily have the joint density of (X,Y).
We then can figure out P(X+Y<z) for z in (0,2) and then (2,4)
where the integration is done over the 2 by 2 square
Now be aware, in the second case, it is smarter to use the complement.
And note that X+Y=c is a line cutting through that square.