91 lines
1.4 KiB
Markdown
91 lines
1.4 KiB
Markdown
---
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title: Randomness tests of a non-uniform distribution
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date: \today
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author:
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- Giulia Marcer
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- Michele Guerini Rocco
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institute:
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- Università di Milano-Bicocca
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theme: metropolis
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aspectratio: 169
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fontsize: 14pt
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mathfont: FiraMath-Regular
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sansfont: Fira Sans
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header-includes: |
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```{=latex}
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% Misc
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% "thus" in formulas
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\DeclareMathOperator{\thus}{%
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\hspace{30pt} \Longrightarrow \hspace{30pt}
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}
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% "et" in formulas
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\DeclareMathOperator{\et}{%
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\hspace{30pt} \wedge \hspace{30pt}
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}
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```
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...
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# Goal
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## Goal
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What?
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- Generate a sample of points from a Moyal PDF
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- Prove it truly comes from it and not from a Landau PDF
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How?
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- Applying some hypothesis testings
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Why?
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- They are really similar. Historically, the Moyal distribution was utilized in
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the approximation of the Landau Distribution.
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# Two similar distributions
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:::: {.columns .c}
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::: {.column width=50%}
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\begin{center}
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Landau PDF
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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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\end{center}
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:::
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::: {.column width=50%}
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\begin{center}
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Moyal PDF
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$$
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M(x) = \frac{1}{\sqrt{2 \pi \sigma}} \exp \left( - \frac{x - \mu }{2 \sigma}
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- \frac{1}{2} e^{- \frac{x -\mu}{\sigma}} \right)
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$$
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\end{center}
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:::
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::::
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:::: {.columns .c}
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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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## Two similar distributions
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grafici sovrapposti
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