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?