Moment Generating Function of Geometric Distribution/Formulation 2/Examples/First Moment

Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 2

Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$

$\map \Pr {X = k} = p \paren {1 - p}^k$

The first moment generating function of $X$ is given by:

$\map { {M_X}'} t = \dfrac {p \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^2 }$

Proof

We have:

\(\ds \map { {M_X}'} t\)

\(=\)

\(\ds \map {\frac \d {\d t} } {\dfrac p {1 - \paren {1 - p} e^t} }\)

Moment Generating Function of Geometric Distribution

\(\ds \)

\(=\)

\(\ds \dfrac {-p \paren {-\paren {1 - p} e^t } } {\paren {1 - \paren {1 - p} e^t}^2 }\)

Quotient Rule for Derivatives, Derivative of Exponential Function

\(\ds \)

\(=\)

\(\ds \dfrac {p \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^2 }\)

$\blacksquare$