95 lines
1.5 KiB
Markdown
95 lines
1.5 KiB
Markdown
# MC simulations
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## In summary
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-----------------------------------------------------
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Landau Moyal
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----------------- ----------------- -----------------
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median $m_L\ex$ $m_M\ex (μ, σ)$
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mode $\mu_L\ex$ $\mu_M\ex (μ)$
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FWHM $w_L\ex$ $w_M\ex (σ)$
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-----------------------------------------------------
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## Moyal parameters
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A $M(x)$ similar to $L(x)$ can be found by imposing:
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\vspace{15pt}
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- equal mode
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$$
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\mu_M\ex = \mu_L\ex \approx −0.22278298...
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$$
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. . .
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- equal width
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$$
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w_M\ex = w_L\ex = \sigma \cdot a
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$$
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$$
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\implies \sigma_M \approx 1.1191486...
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$$
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## Moyal parameters
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:::: {.columns}
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::: {.column width=50%}
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:::
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::: {.column width=50%}
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:::
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::::
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## Moyal parameters
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This leads to more different medians:
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\begin{align*}
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m_M = 0.787... \thus &m_M = 0.658... \\
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&m_L = 1.355...
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\end{align*}
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## Landau Sample
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Sample N random points following $L(x)$
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$$
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L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
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dt \, e^{-t \ln(t) -xt} \sin (\pi t)
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$$
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. . .
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gsl_ran_Landau(gsl_rng)
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## Moyal sample
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Sample N random points following $M_{\mu \sigma}(x)$
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$$
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M_{\mu \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
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\left[ - \frac{1}{2} \left(
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\frac{x - \mu}{\sigma}
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+ e^{-\frac{x - \mu}{\sigma}} \right) \right]
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$$
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. . .
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reverse sampling
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- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$
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