# Landau distribution


## Landau PDF

:::: {.columns align=center}
:::  {.column width=50%}
  $$
    L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
           dt \, e^{-t \ln(t) -xt} \sin (\pi t)
  $$
:::

:::  {.column width=50%}
  ![](images/landau-pdf.pdf)
:::
::::

  . . .

\begin{center}
  No closed form for \alert{ANYTHING}
\end{center}


## Landau median

::::: {.columns}

:::: {.column width=50%}

::: incremental
  - The median of $f$ is defined by
  $$
    F(m) = \int_{-\infty}^m fdx = \frac{1}{2}
  $$

  - Equivalently
    $$
    m = F^{-1}\left(\frac{1}{2}\right)
   $$

  -  PDF Numerical integration up to $1/2$ or QDF is needed
:::

::::

::: {.column width=50%}
  ![](images/median.pdf)
:::

:::::


## Landau median

- CDF computed by numerical integration
- Median computed by numerical root-finding
$$
  F(x) = \frac{1}{2} \thus m_L\ex = 1.3557804...
$$

\setbeamercovered{}
\begin{center}
  \begin{tikzpicture}[overlay]
    \pause
    \node [opacity=0.5, xscale=0.35, yscale=0.25 ] at (2.4,0.95) {\includegraphics{images/high.png}};
  \end{tikzpicture}
\end{center}

\setbeamercovered{transparent}


## Landau mode

::::: {.columns}

:::: {.column width=50%}

  - Maximum $\hence \partial_x L(\mu) = 0$

  . . .

  \vspace{20pt}

  - 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=0.65, yscale=0.5 ] at (here)
            {\includegraphics{images/high.png}};
    \end{tikzpicture}
  \end{center}

  \setbeamercovered{transparent}

::::

::: {.column width=50%}
  ![](images/mode.pdf)
:::
:::::


## Landau FWHM

::::: {.columns}

:::: {.column width=50%}
  $$
    \text{FWHM} = w = x_+ - x_-
  $$
  $$
    L(x_{\pm}) = \frac{L_{\text{max}}}{2} = \frac{L(\mu_L)}{2}
  $$

  . . .

  \vspace{20pt}

  - Computed by numerical root finding (Brent)

  \setbeamercovered{}

  \begin{center}
    \begin{tikzpicture}[remember picture]
      \node at (-1,0) (here) {$w_L\ex = 4.018645...$};
      \pause
      \node [opacity=0.5, xscale=0.65, yscale=0.5 ] at (here)
            {\includegraphics{images/high.png}};
    \end{tikzpicture}
  \end{center}

  \setbeamercovered{transparent}
::::

::: {.column width=50%}
  ![](images/fwhm.pdf)
:::

:::::