# PMF for the sum of random variables

For a sum of two independent uniform discrete random variables, Z = X + Y, what is the probability mass function of Z? X and Y both take on values between 1 and L

I know that for the sum of independent rv's the PMF is a convolution
so...
Ʃ(1/k)(1/n-k) from k = 1 to L
but i'm wondering.. can this be simplified?

## Answers and Replies

lanedance
Homework Helper
these are good to do geometrically, consider the xy plane, we have LxL grid of discrete (X,Y) outcomes, each equi-probable, with probability 1/L^2

Lines of constant Z=z have a slope of -x, and the probability of Z will be the number of discrete points intersected by a constant

try drawing it, this should also help understand the analytic convolution method

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
No wonder you are having trouble: you are 100% wrong in what you are writing. Go back and apply known results correctly.

RGV

No wonder you are having trouble: you are 100% wrong in what you are writing. Go back and apply known results correctly.

RGV

For a sum of two independent uniform discrete random variables, Z = X + Y, what is the probability mass function of Z? X and Y both take on values between 1 and n

I know that for the sum of independent rv's the PMF is a convolution
so...
Ʃ(1/k)(1/n-k) from k = 1 to n

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
For a sum of two independent uniform discrete random variables, Z = X + Y, what is the probability mass function of Z? X and Y both take on values between 1 and n

I know that for the sum of independent rv's the PMF is a convolution
so...
Ʃ(1/k)(1/n-k) from k = 1 to n

That is not the required convolution. I have no idea what it is.

RGV