# PMF for the sum of random variables

• magnifik
In summary, the conversation is about finding the probability mass function of Z, which is the sum of two independent uniform discrete random variables, X and Y. The PMF is a convolution, but the provided formula is incorrect. The correct formula is still unknown.

#### magnifik

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?

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

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

RGV

Ray Vickson said:
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

magnifik said:
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