# MC simulations


## In summary

-----------------------------------------------------
                  Landau            Moyal
----------------- ----------------- -----------------
median             $m_L\ex$          $m_M\ex (μ, σ)$

mode               $\mu_L\ex$        $\mu_M\ex (μ)$

FWHM               $w_L\ex$          $w_M\ex (σ)$
-----------------------------------------------------


## Moyal parameters

A $M(x)$ similar to $L(x)$ can be found by imposing:

\vspace{15pt}

- equal mode
$$
  \mu_M\ex = \mu_L\ex \thus \mu \approx −0.22278298...
$$

. . .

- equal width
$$
  w_M\ex = w_L\ex = \sigma \cdot a \thus \sigma \approx 1.1191486...
$$


## Moyal parameters

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  ![](images/both-pdf-bef.pdf)
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  ![](images/both-pdf-aft.pdf)
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## Moyal parameters

This leads to more different medians:

\begin{align*}
  m_M = 0.787... \thus &m_M = 0.658... \\
                       &m_L = 1.355...
\end{align*}


## Landau Sample

Sample N random points following $L(x)$

$$
  L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
         dt \, e^{-t \ln(t) -xt} \sin (\pi t)
$$

. . .

gsl_ran_Landau(gsl_rng)


## Moyal sample

Sample N random points following $M_{\mu \sigma}(x)$

$$
  M_{\mu \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
         \left[ - \frac{1}{2} \left(
           \frac{x - \mu}{\sigma}
           + e^{-\frac{x - \mu}{\sigma}} \right) \right]
$$

. . .

reverse sampling

- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$


