Slides10-2010 - \documentclasscfw_beamer...

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\documentclass{beamer}  \usepackage{beamerthemesplit}  \usetheme{Boadilla}  \usecolortheme{albatross}  \title[Moment Generating Functions]{Moment Generating Functions: Lecture X}  \author{Charles B. Moss}  \date{\today}  \begin{document}  \frame{\titlepage}  \section[Outline]{}  \frame{\tableofcontents}  \section{Moment Generating Function}  \frame    \frametitle{Moment Generating Function}    \begin{itemize}    \item {\bf Definition 2.3.3.} Let $X$ be a random variable with cumulative distribution  function $F\left(X\right)$. The moment generating function (mgf) of $X$ (or  $F\left(X\right)$ ), denoted $M_X\left(t\right)$, is  \begin{equation}  M_X\left(t\right) = {\rm E} \left[ e^{tX} \right]  \label{eqn:lect10-001}  \end{equation}    \noindent provided that the expectation exists for $t$ in some neighborhood of 0.  That  is, there is an $h>0$ such that, for all $t$ in $-h < t < h$ , ${\rm E} \left[ e^{tX} \right]$  exists.       \begin{itemize}       \item If the expectation does not exist in a neighborhood of 0, we say that the 
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moment generating function does not exist.       \item More explicitly, the moment generating function can be defined as  \begin{equation}  \begin{array}{c}  \displaystyle M_X \left(t\right) = \int_{-\infty}^\infty e^{tx} f\left(x\right) dx \,{\rm for \,  continuous \, random \, variables, \, and} \\ \\  \displaystyle M_x \left(t\right) = \sum_x e^{tx} P\left[X = x\right] \, {\rm for \, discrete \,  random \, variables}  \end{array}  \label{eqn:lect10-002}  \end{equation}       \end{itemize}    \end{itemize}  \frame    \begin{itemize}    \item {\bf Theorem 2.3.2} If $X$ has {\it mgf} $M_X\left(t\right)$ , then  \begin{equation}  {\rm E} \left[ X^n\right] = M_x^{\left(n\right)}\left(0\right)  \label{eqn:lect10-003}  \end{equation}    \noindent where we define      \begin{equation}  M_X^{\left(n\right)} \left( 0 \right) = \frac{\displaystyle d^n}{\displaystyle d t^n} \left. M_X  \left(t\right) \right|_{t\rightarrow 0}  \label{eqn:lect10-004}  \end{equation}       \begin{itemize}       \item[1.] First note that $e^{tX}$ can be approximated around zero using a Taylor 
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series expansion         \begin{equation}  \begin{array}{c}  \displaystyle M_X \left(t\right) = {\rm E} \left[ e^{tx}\right] = {\rm E}\left[ e^0 + t e^{t0}  \left(x - 0\right) + \right. \\  \displaystyle \left. \frac{1}{2} t^2 e^{t0} \left(x-0\right)^2 + \frac{1}{6} t^3 e^{t0} \left(x- 0\right)^3 + \cdots \right] \\ \\  \displaystyle \,\,\, = 1 + {\rm E}\left[x\right]t+{\rm E}\left[x^2\right]\frac{\displaystyle t^2} {\displaystyle 2} + {\rm E}\left[ x^3 \right] \frac{\displaystyle t^3}{\displaystyle 6} + \cdots 
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Slides10-2010 - \documentclasscfw_beamer...

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