44 lines
1.2 KiB
Markdown
44 lines
1.2 KiB
Markdown
# PDF
|
|
|
|
The Moyal distribution is defined as:
|
|
$$
|
|
M(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} \left[ x + e^{-x} \right]}
|
|
$$
|
|
More generally, it is defined with the location and scale parameters $\mu$ and
|
|
$\sigma$ such as:
|
|
$$
|
|
x \rightarrow \frac{x - \mu}{\sigma}
|
|
$$
|
|
|
|
# CDF
|
|
|
|
The cumulative distribution function $\mathscr{M}(x)$ can be derived from the
|
|
pdf $M(x)$ integrating:
|
|
$$
|
|
\mathscr{M}(x) = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, M(y)
|
|
= \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, e^{- \frac{1}{2}}
|
|
e^{- \frac{1}{2} e^{-y}}
|
|
$$
|
|
with the change of variable:
|
|
\begin{align}
|
|
z = \frac{1}{\sqrt{2}} e^{-\frac{y}{2}}
|
|
&\thus \frac{dz}{dy} = \frac{-1}{2 \sqrt{2}} e^{-\frac{y}{2}} \\
|
|
&\thus dy = -2 \sqrt{2} e^{\frac{y}{2}} dz
|
|
\end{align}
|
|
hence, the limits of the integral become:
|
|
\begin{align}
|
|
y \rightarrow - \infty &\thus z \rightarrow + \infty \\
|
|
y = x &\thus z = \\\frac{1}{\sqrt{2}} e^{-\frac{x}{2}} = f(x)
|
|
\end{align}
|
|
and the CDF can be rewritten as:
|
|
$$
|
|
\mathscr{M}(x) = \frac{1}{2 \pi} \int\limits_{+ \infty}^{f(x)}
|
|
dz \, (- 2 \sqrt{2}) e^{\frac{y}{2}} e^{- \frac{y}{2}} e^{- z^2}
|
|
= \frac{-2 \sqrt{2}}{\sqrt{2 \pi}} \int\limits_{+ \infty}^{f(x)}
|
|
dz e^{- z^2}
|
|
$$
|
|
since the `erf` is defines as:
|
|
$$
|
|
|
|
$$
|