35 lines
1.1 KiB
Markdown
35 lines
1.1 KiB
Markdown
The Moyal distribution, which is a steepest descent approximation of the
|
|
Landau distribition, is defines as:
|
|
$$
|
|
\exp \left( - \frac{x - \mu }{2 \sigma}
|
|
- \frac{1}{2} \exp \left( - \frac{x -\mu}{\sigma} \right) \right)
|
|
$$
|
|
Mean $m$ and variance $\sigma$:
|
|
$$
|
|
m = \mu + \sigma [ \gamma + \ln(2) ] \et \sigma = \frac{\pi^2 \sigma^2}{2}
|
|
$$
|
|
Median:
|
|
$$
|
|
\mu - \sigma \left[ 2 \text{erf}^{-1} \left( \frac{1}{2} \right)^2 \right]
|
|
$$
|
|
skewness and kurtosis are constant:
|
|
$$
|
|
s = \frac{28 \sqrt{2} Z(3)]{\pi^3} \et k = 7
|
|
$$
|
|
max value:
|
|
$$
|
|
\frac{1}{\sqrt{2 e \pi}}
|
|
$$
|
|
cdf:
|
|
$$
|
|
\text{erf} \left( \frac{\exp \left(
|
|
- \frac{x - \mu}{2 \sigma} \right)}{\sqrt{2}} \right)
|
|
$$
|
|
|
|
$\mu$ is the location parameter and $\sigma$ is the scale parameter.
|
|
The Moyal distribution was first proposed in a 1955 paper by physicist J. E.
|
|
Moyal. The distribution models the energy lost by a fast charged particle
|
|
(and hence the number of ion pairs produced) during ionization. Historically,
|
|
the Moyal distribution has been utilized in the approximation of the Landau
|
|
Distribution and has since found use in modeling a wide array of phenomena.
|