129 lines
2.3 KiB
Markdown
129 lines
2.3 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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themeoptions:
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- titleformat=allcaps
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aspectratio: 169
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fontsize: 12pt
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mainfont: Fira Sans
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mainfontoptions:
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- BoldFont=Fira Sans
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mathfont: FiraMath-Regular
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header-includes: |
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```{=latex}
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%% Colors
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\definecolor{mDarkTeal} {HTML}{020202}
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\definecolor{mLightBrown}{HTML}{C49D4A}
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\definecolor{mDarkRed} {HTML}{92182B}
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\definecolor{green} {HTML}{60AC39}
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\definecolor{red} {HTML}{D73737}
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\definecolor{blue} {HTML}{6684E1}
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\definecolor{yellow}{HTML}{CFB017}
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\setbeamercolor{frametitle}{bg=mDarkRed}
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% center images
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\LetLtxMacro{\oldIncludegraphics}{\includegraphics}
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\renewcommand{\includegraphics}[2][]{
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\centering
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\oldIncludegraphics[#1]{#2}
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}
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%% customer macros
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\DeclareMathOperator{\with}{%
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\hspace{30pt} \text{with} \hspace{30pt}
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}
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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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The Landau and Moyal PDFs are really similar. Historically, the latter distribution was utilized in
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the approximation of the Landau Distribution.
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:::: {.columns}
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::: {.column width=33%}
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\centering
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{height=130pt}
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:::
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::: {.column width=33%}
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\centering
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{height=130pt}
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:::
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::: {.column width=33%}
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\centering
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{height=130pt}
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:::
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::::
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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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:::
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::: {.column width=50%}
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\begin{center}
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Moyal PDF
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$$
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M(x) = \frac{1}{\sqrt{2 \pi}} \exp \left[ - \frac{1}{2}
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\left( x + e^{- x} \right) \right]
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$$
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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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\centering
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