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

Examples of Use of Moment Generating Function of Geometric Distribution

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 second moment generating function of $X$ is given by:

$\map { {M_X}} t = p \paren {1 - p} e^t \paren {\dfrac {1 + \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^3 } }$

Proof

We have:

\(\ds \map { {M_X}} t\)

\(=\)

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

Definition of Moment Generating Function

\(\ds \)

\(=\)

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

Moment Generating Function of Geometric Distribution: First Moment

\(\ds \)

\(=\)

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

factoring out the $p \paren {1 - p}$

\(\ds \)

\(=\)

\(\ds p \paren {1 - p} \paren {\paren {e^t } \paren {\paren {1 - \paren {1 - p} e^t}^{-2} } + \paren {e^t } \paren {-2 \paren {1 - \paren {1 - p} e^t}^{-3} } \paren {-\paren {1 - p} e^t} }\)

Product Rule for Derivatives, Chain Rule for Derivatives, Derivative of Power, Derivative of Exponential Function

\(\ds \)

\(=\)

\(\ds p \paren {1 - p} \paren {\dfrac {\paren {e^t } } {\paren {1 - \paren {1 - p} e^t}^2 } \dfrac {\paren {1 - \paren {1 - p} e^t} } {\paren {1 - \paren {1 - p} e^t} } + \dfrac {2 \paren {1 - p} e^{2t} } {\paren {1 - \paren {1 - p} e^t}^3 } }\)

multiplying by $1$

\(\ds \)

\(=\)

\(\ds p \paren {1 - p} \paren {\dfrac {e^t - \paren {1 - p} e^{2t} + 2 \paren {1 - p} e^{2t} } {\paren {1 - \paren {1 - p} e^t}^3 } }\)

simplifying

\(\ds \)

\(=\)

\(\ds p \paren {1 - p} e^t \paren {\dfrac {1 + \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^3 } }\)

$\blacksquare$