analistica/slides/sections/1.md

Goal

Goal

Construct six statistical tests to assert whether a sample comes from a Landau distribution

. . .

• Generate a sample L from a Landau PDF
• Generate a sample M from a Moyal PDF

. . .

H_0: sample following Landau PDF

• can we accept H_0 for L?
• can we reject H_0 for M?

Why Moyal?

The Landau and Moyal PDFs are really similar. Historically, the latter was utilized as an approximation of the former.

\includegraphics<1>[height=5.5cm]{images/moyal-photo.jpg} \includegraphics<2>[height=5.5cm]{images/mondau-photo.jpg} \includegraphics<3>[height=5.5cm]{images/landau-photo.jpg}

Two similar distributions

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

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

:::

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

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

::: ::::

\vspace{1em}

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

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

Two similar distributions

Statistical tests

. . .

• Properties test

  compatibility between expected and observed PDF properties

. . .

• Kolmogorov - Smirnov test

  compatibility between expected and empirical CDF

. . .

• Trapani test

  test for finite or infinite moments