Moment Generating Function of Discrete Uniform Distribution
Jump to navigation
Jump to search
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:
- $\ds \map {M_X} t = \expect {e^{t X} } = \sum_{N \mathop = 1}^n \map \Pr {X = N} e^{N t}$
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 $r = e^t$ |
$\blacksquare$