# Distribution for Arithmetic Mean of n Geometrically Distributed Random Variables

For the evaluation of an algorithm I implemented for work, I need to find the distribution function for the arithmetic mean of $n$ independent, geometrically distributed random variables.

Let $X_1,X_2,...,X_n$ be geometrically distributed random values, such that $Pr(X_i=x) = (\frac{1}{2})^{x}$ with $x = 1,2,\dots$. Let $Z$ be a random variable for the arithmetic mean of $X_1,...,X_n$.

$Z = \frac{1}{n}\sum_{i=1}^{n}X_i$

What is $Pr(Z=x)$?

Thanks a lot for your help.

• Here's a question that is similar to yours. The arithmetic mean will have a negative binomial distribution. – Kitegi May 17 '15 at 16:25