/sci/ - Science & Math

Hey /sci/ help a mathgeek out

So I've got some iids of natural numbers, I'm going to call them <span class="math">X_i[/spoiler] for i=1,...,n.

If I define <span class="math">X = \sum_i X_i[/spoiler], is there an easy way to compute <span class="math">P(X=m)[/spoiler]?

The naive way I'm doing it at the moment requires me to manually count the partitions; for example, if n=4 and m=3 then the sum works out as

<span class="math">P(X=3) = {}^4C_1 P(X_i=3) P(X_i=0)^3
+ {}^4C_2{}^2C_1 P(X_i=2) P(X_i=1) P(X_i=0)^2
+ {}^4C_3 P(X_i=1)^3 P(X_i=0)[/spoiler]

As I'm counting over the different partitions of m, it's very hard to generalize. I want to be able to have m and n go up to about 50.

More context in comments.