1. ## Moment Generating Function

Hi. I have a problem with this question:

Let $\displaystyle X$ be a discrete uniform random variable with pmf $\displaystyle p_{X}(x)=\frac{1}{N}$, for $\displaystyle x=1,2,...,N$. Find the mgf of $\displaystyle X$.

What I did was:

$\displaystyle M_{X}(t)=E[e^{tx}]$
$\displaystyle =\sum_{x=1}^N e^{tx}\frac{1}{N}$
$\displaystyle =\frac{1}{N} \sum_{x=1}^N e^{tx}$

I don't know how to continue after that

2. Originally Posted by Zayyym
Hi. I have a problem with this question:

Let $\displaystyle X$ be a discrete uniform random variable with pmf $\displaystyle p_{X}(x)=\frac{1}{N}$, for $\displaystyle x=1,2,...,N$. Find the mgf of $\displaystyle X$.

What I did was:

$\displaystyle M_{X}(t)=E[e^{tx}]$
$\displaystyle =\sum_{x=1}^N e^{tx}\frac{1}{N}$
$\displaystyle =\frac{1}{N} \sum_{x=1}^N e^{tx}$

I don't know how to continue after that
$\displaystyle \sum_{x=1}^N e^{tx} = \sum_{x=1}^N (e^{t})^x$

is a geometric series. The first term is $\displaystyle e^t$ and the ratio is $\displaystyle e^t$.

If you scroll about two thirds of the way down this thread: Uniform Distribution -- from Wolfram MathWorld

you'll find the answer you should get.