Distributions and Statistics
The Gaussian
\begin{equation} g(x)=e^{\left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)} \end{equation}
$\sigma = 1$
\begin{equation} g(x)=e^{\left(-{\frac {(x-X)^{2}}{2\sigma ^{2}}}\right)} \end{equation}
Offset in $x$
Normalize it
\begin{equation} \int_{-\infty}^{\infty} f(x) dx = 1 \end{equation}
\begin{equation} f(x) = N e^{\left(-{\frac {(x-X)^{2}}{2\sigma ^{2}}}\right)} \end{equation} \begin{equation} \int_{-\infty}^{\infty} f(x) dx = \int_{-\infty}^{\infty} N e^{\left(-{\frac {(x-X)^{2}}{2\sigma ^{2}}}\right)} dx \end{equation}
\begin{equation} \int_{-\infty}^{\infty} f(x) dx = N \sigma \sqrt{2\pi} \end{equation}
\begin{equation} N = \frac{1}{\sigma \sqrt{2 \pi}} \end{equation}
Normalized Gaussian Distribution
\begin{equation} G_{X,\sigma}(x)=\frac{1}{\sigma\sqrt{2 \pi}}e^{\left(-{\frac {(x-X)^{2}}{2\sigma ^{2}}}\right)} \end{equation}
This distribution applys when the measurement is subject to random errors.
