68 lines
860 B
Markdown
68 lines
860 B
Markdown
# Landau PDF
|
||
|
||
|
||
## A pathological distribution
|
||
|
||
Because of its fat tail:
|
||
|
||
\begin{align*}
|
||
E[x] &\longrightarrow + \infty \\
|
||
V[x] &\longrightarrow + \infty
|
||
\end{align*}
|
||
|
||
. . .
|
||
|
||
No closed form for parameters $\thus$ numerical estimations
|
||
|
||
|
||
## Landau median
|
||
|
||
The median of a PDF is defined as:
|
||
|
||
$$
|
||
m = Q \left( \frac{1}{2} \right)
|
||
$$
|
||
|
||
. . .
|
||
|
||
- CDF computed by numerical integration
|
||
- QDF computed by numerical root-finding (Brent)
|
||
|
||
$$
|
||
m_L\ex = 1.3557804...
|
||
$$
|
||
|
||
|
||
## Landau mode
|
||
|
||
- Maximum $\hence \partial_x L(\mu) = 0$
|
||
|
||
. . .
|
||
|
||
- Computed by numerical minimization (Brent)
|
||
|
||
$$
|
||
\mu_L\ex = − 0.22278...
|
||
$$
|
||
|
||
|
||
## Landau FWHM
|
||
|
||
We need to compute the maximum:
|
||
|
||
$$
|
||
L_{\text{max}} = L(\mu_L)
|
||
$$
|
||
|
||
$$
|
||
\text{FWHM} = w = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
|
||
$$
|
||
|
||
. . .
|
||
|
||
- Computed by numerical root finding (Brent)
|
||
|
||
$$
|
||
w_L\ex = 4.018645...
|
||
$$
|