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.
