1.5 KiB
1.5 KiB
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 \approx −0.22278298...
. . .
- equal width
w_M\ex = w_L\ex = \sigma \cdot a
\implies \sigma_M \approx 1.1191486...
Moyal parameters
:::: {.columns}
::: {.column width=50%}
::: {.column width=50%}
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
yuniformly in [0, 1]\hence x = Q_M(y)