# What is the distribution of the number of occurrences of a given integer, in repeated samples drawn from Uniform Integer [1, N] distribution?

If I draw M repeated samples from Uniform Integer Distribution on [1,N], what will be the distribution of the number of times a given integer chosen among the integers 1,2,...,N appears in my M samples?

For concreteness, lets say that I draw repeatedly 100 times from Uniform Integer [1,5] distribution. What then will be the distribution of the number of times the integer 3 appears in my 100 samples?

## 1 Answer

In a single draw, your chosen integer will be drawn with probability $$\frac{1}{N}$$. Since you draw with replacement, the total number of such successes is binomially distributed with parameters $$M$$ and $$p=\frac{1}{N}$$.