analistica/slides/sections/8.md

Landau sample

Sample

Sample N = 50'000 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)

Compatibility results:

Median:

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

• t = 0.761
• $p = 0.446$ :::

::: {.column width=50%}

$$
\hence \text{Compatible!}

::: ::::

\vspace{10pt}

. . .

Mode:

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

• t = 1.012
• $p = 0.311$ :::

::: {.column width=50%}

$$
\hence \text{Compatible!}

::: ::::

\vspace{10pt}

. . .

FWHM:

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

• t=1.338
• $p=0.181$ :::

::: {.column width=50%}

$$
\hence \text{Compatible!}

::: ::::

Moyal sample

Sample

Sample N = 50'000 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)

Compatibility results:

Median:

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

• t = 669.940
• $p = 0.000$ :::

::: {.column width=50%}

$$
\hence \text{Not compatible!}

::: ::::

\vspace{10pt}

. . .

Mode:

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

• t = 0.732
• $p = 0.464$ :::

::: {.column width=50%}

$$
\hence \text{Compatible!}

::: ::::

\vspace{10pt}

. . .

FWHM:

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

• t = 1.329
• $p = 0.184$ :::

::: {.column width=50%}

$$
\hence \text{Compatible!}

::: ::::

KS results

Samples results

N = 50000 sampled points

. . .

Landau sample:

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

• D = 0.004
• $p = 0.379$ :::

::: {.column width=50%}

$$
\hence \text{Compatible!}

::: ::::

\vspace{10pt}

. . .

Moyal sample:

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

• D = 0.153
• $p = 0.000$ :::

::: {.column width=50%}

$$
\hence \text{Not compatible!}

::: ::::

Trapani results

Samples results

. . .

Landau sample:

:::: {.columns} ::: {.column width=33%}

$$
\mu_1
\begin{cases}
  \Theta = 0.255 \\
  p = 0.614
\end{cases}

:::

::: {.column width=33% .c}

$$
\mu_2
\begin{cases}
  \Theta = 0.432 \\
  p = 0.511
\end{cases}

:::

::: {.column width=33% .c}

$$
\hence \text{Infinite!}

::: ::::

. . .

\vspace{20pt}

Moyal sample:

:::: {.columns} ::: {.column width=33%}

$$
\mu_1
\begin{cases}
  \Theta^2 = 106 \\
  p = 0.000
\end{cases}

:::

::: {.column width=33%}

$$
\mu_2
\begin{cases}
  \Theta^2 = 162 \\
  p = 0.000
\end{cases}

:::

::: {.column width=33% .c}

$$
\hence \text{Finite!}

::: ::::