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Section 9.5 Generating Functions for Normal and Associated Distributions
Theorem 9.5.1. Moment Generating Function for Normal.
Presuming \(t \gt 0\) and
\begin{equation*}
M(t) = e^{t \mu+\frac{1}{2}t^2\sigma^2}
\end{equation*}
Proof.
\begin{align*}
M(t) & = \int_{-\infty}^{\infty} e^{tx} \frac{1}{\sigma \sqrt{2 \pi}} e^{ -\left ( \frac{x-\mu}{\sigma} \right ) ^2 / 2} dx\\
& = \int_{-\infty}^{\infty} e^{t(z \sigma + \mu)} \frac{1}{\sqrt{2 \pi}} e^{ -z ^2 / 2} dz\\
& = e^{t \mu} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -(z ^2 - 2t z \sigma ) / 2} dz\\
& = = e^{t \mu} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -(z ^2 - 2t z \sigma + t^2 \sigma^2 - t^2 \sigma^2 ) / 2} dz\\
& = e^{t \mu+\frac{1}{2}t^2\sigma^2} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -\left (z - t \sigma \right )^2 / 2} dz\\
& = e^{t \mu+\frac{1}{2}t^2\sigma^2} \cdot 1
\end{align*}
where the final integral is just a shifted standard normal and therefore has value 1.
Corollary 9.5.2. Normal Properties via Moment Generating Function.
For the Normal variable X,
\begin{equation*}
M(0) = 1
\end{equation*}
\begin{equation*}
M'(0) = \mu
\end{equation*}
\begin{equation*}
M''(0) = \sigma^2 + \mu^2
\end{equation*}
Proof.
\begin{equation*}
M(0) = e^{0 \mu+\frac{1}{2}0^2\sigma^2} = e^0 = 1.
\end{equation*}
Continuing,
\begin{equation*}
M'(t) = {\left(\sigma^2 t + \mu \right)} e^{\left(\frac{1}{2} \sigma^2 t^{2} + \mu t \right)}
\end{equation*}
and therefore
\begin{equation*}
M'(0) = {\left(\sigma^2 0 + \mu \right)} e^{\left(\frac{1}{2} \sigma^2 0^{2} + \mu 0 \right)} = \mu e^0 = \mu.
\end{equation*}
Continuing with the second derivative,
\begin{equation*}
M''(t) = {\left(\sigma^2 t + \mu\right)}^2 e^{\left(\frac{1}{2} \sigma^2 t^2 + \mu t\right)} + \sigma^2 e^{\left(\frac{1}{2} \sigma^2 t^2 + \mu t\right)}
\end{equation*}
and therefore
\begin{equation*}
M''(0) = {\left(\sigma^2 0 + \mu\right)}^2 e^{\left(\frac{1}{2} \sigma^2 0^2 + \mu 0\right)} + \sigma^2 e^{\left(\frac{1}{2} \sigma^2 0^2 + \mu 0\right)} = \mu^2 + \sigma^2
\end{equation*}
which is the squared mean plus the variance for the normal distribution.
Theorem 9.5.3. Moment Generating Function for Chi-Square.
Presuming \(t \gt 0\) and
\begin{equation*}
M(t) = \left ( \frac{1}{1-2t} \right )^{r/2}
\end{equation*}
Proof.
\begin{align*}
M(t) & = \int_0^{\infty} e^{tx} \frac{x^{r/2-1} e^{-x/2} }{\Gamma(r/2) 2^{r/2} } dx\\
& = \int_0^{\infty} \frac{x^{r/2-1} e^{-x(1-2 t)/2} }{\Gamma(r/2) 2^{r/2}} dx\\
& = \int_0^{\infty} \frac{\left ( \frac{u}{1-2t} \right )^{r/2-1} e^{-u/2} }{\Gamma(r/2) 2^{r/2} } \frac{1}{1-2t} du\\
& = \left ( \frac{1}{1-2t} \right )^{r/2} \int_0^{\infty} \frac{u^{r/2-1} e^{-u/2} }{\Gamma(r/2) 2^{r/2}} du\\
& = \left ( \frac{1}{1-2t} \right )^{r/2}
\end{align*}
where the final integral is again Chi-Square and therefore has value 1.
Corollary 9.5.4. Chi-Square Properties via Moment Generating Function.
For the Chi-Square variable X,
\begin{equation*}
M(0) = 1
\end{equation*}
\begin{equation*}
M'(0) = r = \mu
\end{equation*}
\begin{equation*}
M''(0) = 2r + r^2 = \sigma^2 + \mu^2
\end{equation*}
Proof.
\begin{equation*}
M(0) = \left ( \frac{1}{1-2 \cdot 0} \right )^{r/2} = 1.
\end{equation*}
Continuing,
\begin{equation*}
M'(t) = \frac{r {\left(-2 t + 1\right)}^{\frac{1}{2} r - 1}}{{\left(-2 t + 1 \right)}^{r}}
\end{equation*}
and therefore
\begin{equation*}
M'(0) = \frac{r {\left(-2 \cdot 0 + 1\right)}^{\frac{1}{2} r - 1}}{{\left(-2 \cdot 0 + 1 \right)}^{r}} = \frac{r}{1} = r.
\end{equation*}
Continuing with the second derivative,
\begin{equation*}
M''(t) = -\frac{{\left(r - 2\right)} r {\left(-2 t + 1\right)}^{\frac{1}{2} r - 2}}{{\left(-2 t + 1\right)}^{r}} + \frac{2 r^{2} {\left(-2 t + 1\right)}^{r - 2}}{{\left(-2 t + 1\right)}^{\frac{3}{2} r}}
\end{equation*}
and therefore
\begin{align*}
M''(0) & = -\frac{{\left(r - 2\right)} r {\left(-2 \cdot 0 + 1\right)}^{\frac{1}{2} r - 2}}{{\left(-2 \cdot 0 + 1\right)}^{r}} + \frac{2 r^{2} {\left(-2 \cdot 0 + 1\right)}^{r - 2}}{{\left(-2 \cdot 0 + 1\right)}^{\frac{3}{2} r}} \\
& = -\frac{{\left(r - 2\right)} r }{1} + \frac{2 r^{2} }{1}= 2r + r^2 = \sigma^2 + \mu^2
\end{align*}
which is the squared mean plus the variance for the normal distribution.