1.7 KiB
Landau PDF
Landau PDF
:::: {.columns} ::: {.column width=50% align=center}
$$
L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
dt \, e^{-t \ln(t) -xt} \sin (\pi t)
:::
::: {.column width=50%}
. . .
No closed form for \textcolor{cyclamen}{ANYTHING}
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)
\setbeamercovered{}
\begin{center} \begin{tikzpicture}[remember picture] \node at (0,0) (here) {$m_L\ex = 1.3557804...$}; \pause \node [opacity=0.5, xscale=0.35, yscale=0.25 ] at (here) {\includegraphics{images/high.png}}; \end{tikzpicture} \end{center}
\setbeamercovered{transparent}
Landau mode
- Maximum
\hence \partial_x L(\mu) = 0
. . .
- Computed by numerical minimization (Brent)
\setbeamercovered{}
\begin{center} \begin{tikzpicture}[remember picture] \node at (0,0) (here) {$\mu_L\ex = − 0.22278...$}; \pause \node [opacity=0.5, xscale=0.32, yscale=0.25 ] at (here) {\includegraphics{images/high.png}}; \end{tikzpicture} \end{center}
\setbeamercovered{transparent}
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)
\setbeamercovered{}
\begin{center} \begin{tikzpicture}[remember picture] \node at (0,0) (here) {$w_L\ex = 4.018645...$}; \pause \node [opacity=0.5, xscale=0.32, yscale=0.25 ] at (here) {\includegraphics{images/high.png}}; \end{tikzpicture} \end{center}
\setbeamercovered{transparent}