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

From ProofWiki
Jump to navigation Jump to search

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

$\map { {M_X}'''} t = p \paren {1 - p} e^t \paren {\dfrac {1 + 4 \paren {1 - p} e^t + \paren {1 - p}^2 e^{2t} } {\paren {1 - \paren {1 - p} e^t}^4 } }$


Proof

We have:

\(\ds \map { {M_X}'''} t\) \(=\) \(\ds \frac \d {\d t} \map { {M_X}''} t\) Definition of Moment Generating Function
\(\ds \) \(=\) \(\ds \frac \d {\d t} p \paren {1 - p} e^t \paren {\dfrac {1 + \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^3 } }\) Moment Generating Function of Geometric Distribution: Second Moment
\(\ds \) \(=\) \(\ds p \paren {1 - p} \paren {\frac \d {\d t} \paren {\paren { e^t + \paren {1 - p} e^{2t} } \paren {1 - \paren {1 - p} e^t}^{-3} } }\)
\(\ds \) \(=\) \(\ds p \paren {1 - p} \paren {\paren {e^t + 2 \paren {1 - p} e^{2t} } \paren {1 - \paren {1 - p} e^t}^{-3} - 3 \paren {e^t + \paren {1 - p} e^{2t} } \paren {1 - \paren {1 - p} e^t}^{-4} \paren {-\paren {1 - p} e^t} }\) Product Rule, Chain Rule for Derivatives, Derivative of Power, Derivative of Exponential Function
\(\ds \) \(=\) \(\ds p \paren {1 - p} \paren {\dfrac {e^t + 2 \paren {1 - p} e^{2t} } {\paren {1 - \paren {1 - p} e^t}^3 } + \dfrac {3 \paren {1 - p} e^{2t} + 3 \paren {1 - p}^2 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^4 } }\) gathering terms
\(\ds \) \(=\) \(\ds p \paren {1 - p} \paren {\dfrac {e^t + 2 \paren {1 - p} e^{2t} } {\paren {1 - \paren {1 - p} e^t}^3 } \dfrac {\paren {1 - \paren {1 - p} e^t} } {\paren {1 - \paren {1 - p} e^t} } + \dfrac {3 \paren {1 - p} e^{2t} + 3 \paren {1 - p}^2 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^4 } }\) multiplying by $1$
\(\ds \) \(=\) \(\ds p \paren {1 - p} e^t \paren {\dfrac {1 + 2 \paren {1 - p} e^t - \paren {1 - p} e^t - 2 \paren {1 - p}^2 e^{2t} + 3 \paren {1 - p} e^t + 3 \paren {1 - p}^2 e^{2t} } {\paren {1 - \paren {1 - p} e^t}^4 } }\) simplifying
\(\ds \) \(=\) \(\ds p \paren {1 - p} e^t \paren {\dfrac {1 + 4 \paren {1 - p} e^t + \paren {1 - p}^2 e^{2t} } {\paren {1 - \paren {1 - p} e^t}^4 } }\) simplifying

$\blacksquare$