calculating probabilities about the sum of dependent discrete uniforms

Question

Say I have the following information: $$ X_i \sim \text{Discrete Uniform}(1,13) $$ and I want to find $\mathbb P(X_1+X_2+X_3 \ge 25)$ for the cases where the $X_i$'s are dependent. What approximations can be made if we assume they are independent, or is there some distribution that we can use to solve this? I know for continuous uniforms we have the irwin hall, but I couldn't find a discrete uniform version of this?

Answer 1

The sum of discrete uniforms, taken to a quantity big enough is "normal" . Anyway as far as I know , even there's not theoretical distribution for what you're asking you alway can do this:

If you have two variables X1, X2, then your space is all the possible values that X1+X2 can take, let's say that space goes from 1 to 26. Then

for a value Z in that space you have

So every possible value can be calculated.

Also you can approximate results via simulation. For example, in R you can do:

v<-rep(NA,100000)

for (i in (1:100000)) {
  u13 <- sample(1:13, 3, replace = TRUE) 
  su13<-sum(u13)
  v[i]=su13
}
hist(v,breaks=500,xlim=c(3,39))