Moment Generating Function of Discrete Uniform Distribution

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Theorem

Let $X$ be a discrete random variable with a discrete uniform distribution with parameter $n$ for some $n \in \N$.

Then the moment generating function $M_X$ of $X$ is given by:

$\map {M_X} t = \dfrac {e^t \paren {1 - e^{n t} } } {n \paren {1 - e^t} }$


Proof

From the definition of the Discrete Uniform distribution, $X$ has probability mass function:

$\map \Pr {X = N} = \dfrac 1 n$

From the definition of a moment generating function:

$\displaystyle \map {M_X} t = \expect {e^{t X} } = \sum_{N \mathop = 1}^n \map \Pr {X = N} e^{N t}$

So:

\(\ds \map {M_X} t\) \(=\) \(\ds \frac 1 n \sum_{N \mathop = 1}^n \paren {e^t}^N\)
\(\ds \) \(=\) \(\ds \frac {e^t} n \sum_{N \mathop = 0}^{n - 1} \paren {e^t}^N\)
\(\ds \) \(=\) \(\ds \frac {e^t \paren {1 - e^{n t} } } {n \paren {1 - e^t} }\) Sum of Geometric Sequence with $r = e^t$

$\blacksquare$