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joint_dist_X_abs_X_normal_dist.tex
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\author{Habib Ezzatabadi}
\date{$2024$}
\title{"Joint Distribution of $X$ and $|X|$ when $X \sim \mathcal{N}(0, \sigma^2)$}
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\begin{document}
\maketitle
\paragraph{\texorpdfstring{We want to calculate Joint Distribution of
\(X\) and \(|X|\) When
\(X \sim \mathcal{N}(0, \sigma^2)\)}{We want to calculate Joint Distribution of X and \textbar X\textbar{} When X \textbackslash sim \textbackslash mathcal\{N\}(0, \textbackslash sigma\^{}2)}}\label{we-want-to-calculate-joint-distribution-of-x-and-x-when-x-sim-mathcaln0-sigma2}
\[
\begin{aligned}
& X\sim\mathcal{N}(0, \sigma^2), \quad Y = \vert X \vert \implies \\
& P^* = \mathcal{P}(X \leq x, Y \leq y) = F_{X, Y}(x, y) = ? \\
& \mathcal{P}(X \leq x, Y \leq y) = \mathcal{P}(X \leq x, Y \leq y, X > 0) + \mathcal{P}(X \leq x, Y \leq y, X < 0)\\
& \mathcal{P}(0 < X \leq x, ~ X \leq y) + \mathcal{P}(X \leq \min(0, x), ~0 < -X < y) \implies \\
& P^* = \mathcal{P}(0 < X \leq \min(x, y)) + \mathcal{P}(X \leq \min(0, x), -y < X < 0) \implies \\
& P^* = \begin{cases}
x > 0: & \mathcal{P}(-y \leq X \leq \min(x, y)) \\
x < 0: & \begin{cases}
x > -y: & \mathcal{P}(-y < X < x) \\
x < -y: & 0.
\end{cases}
\end{cases}
\end{aligned}
\]
\subsubsection{To show that the way we followed was the correct method.
We also estimate the results based on the simulation and compare it to
the above
proposition.}\label{to-show-that-the-way-we-followed-was-the-correct-method.-we-also-estimate-the-results-based-on-the-simulation-and-compare-it-to-the-above-proposition.}
\texttt{python\ code}
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\prompt{In}{incolor}{66}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{c+c1}{\PYZsh{} load require libraries }
\PY{k+kn}{from} \PY{n+nn}{scipy} \PY{k+kn}{import} \PY{n}{stats}
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{k+kn}{import} \PY{n+nn}{pandas} \PY{k}{as} \PY{n+nn}{pd}
\PY{k}{class} \PY{n+nc}{get\PYZus{}prob}\PY{p}{(}\PY{p}{)}\PY{p}{:}
\PY{k}{def} \PY{n+nf+fm}{\PYZus{}\PYZus{}init\PYZus{}\PYZus{}}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{nsim}\PY{p}{)}\PY{p}{:}
\PY{c+c1}{\PYZsh{}\PYZsh{} define number of simulation}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{nsim} \PY{o}{=} \PY{n}{nsim}
\PY{c+c1}{\PYZsh{} define method for get x and y}
\PY{k}{def} \PY{n+nf}{get\PYZus{}xy}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{x} \PY{o}{=} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{y} \PY{o}{=} \PY{l+m+mi}{2}\PY{p}{,} \PY{n}{sigg} \PY{o}{=} \PY{l+m+mi}{2}\PY{p}{,} \PY{n}{seed} \PY{o}{=} \PY{l+m+mi}{123}\PY{p}{)}\PY{p}{:}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{x} \PY{o}{=} \PY{n}{x}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{y} \PY{o}{=} \PY{n}{y}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{sigg} \PY{o}{=} \PY{n}{sigg}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{seed} \PY{o}{=} \PY{n}{seed}
\PY{c+c1}{\PYZsh{}\PYZsh{} define method for implement simulation }
\PY{k}{def} \PY{n+nf}{sim\PYZus{}main}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
\PY{n}{sim\PYZus{}res} \PY{o}{=} \PY{n}{stats}\PY{o}{.}\PY{n}{norm}\PY{o}{.}\PY{n}{rvs}\PY{p}{(}\PY{n}{loc} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{n}{scale} \PY{o}{=} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{sigg}\PY{p}{,} \PY{n}{size} \PY{o}{=} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{nsim}\PY{p}{,} \PY{n}{random\PYZus{}state} \PY{o}{=} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{seed}\PY{p}{)}
\PY{n}{temp1} \PY{o}{=} \PY{n+nb}{list}\PY{p}{(}\PY{n+nb}{map}\PY{p}{(}\PY{k}{lambda} \PY{n}{u}\PY{p}{:} \PY{l+m+mi}{1} \PY{k}{if} \PY{p}{(}\PY{n}{u} \PY{o}{\PYZlt{}}\PY{o}{=} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{x} \PY{o+ow}{and} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{u}\PY{p}{)} \PY{o}{\PYZlt{}}\PY{o}{=} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{y}\PY{p}{)} \PY{k}{else} \PY{l+m+mi}{0}\PY{p}{,} \PY{n}{sim\PYZus{}res}\PY{p}{)}\PY{p}{)}
\PY{k}{return} \PY{n}{np}\PY{o}{.}\PY{n}{mean}\PY{p}{(}\PY{n}{temp1}\PY{p}{)}
\PY{c+c1}{\PYZsh{}\PYZsh{} define method for exactly value }
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\PY{c+c1}{\PYZsh{} return results }
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\begin{Verbatim}[commandchars=\\\{\}]
nsim x y Sig sim\_result Exact\_result
1 10000 -2 0.5 0.1 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -2 0.5 0.1 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 1.5 0.1 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -2 1.5 0.1 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 2.5 0.1 0.0 2.753624e-89
nsim x y Sig sim\_result Exact\_result
1 100000 -2 2.5 0.1 0.0 2.753624e-89
nsim x y Sig sim\_result Exact\_result
1 10000 -2 3.5 0.1 0.0 2.753624e-89
nsim x y Sig sim\_result Exact\_result
1 100000 -2 3.5 0.1 0.0 2.753624e-89
nsim x y Sig sim\_result Exact\_result
1 10000 -1 0.5 0.1 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -1 0.5 0.1 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -1 1.5 0.1 0.0 7.619853e-24
nsim x y Sig sim\_result Exact\_result
1 100000 -1 1.5 0.1 0.0 7.619853e-24
nsim x y Sig sim\_result Exact\_result
1 10000 -1 2.5 0.1 0.0 7.619853e-24
nsim x y Sig sim\_result Exact\_result
1 100000 -1 2.5 0.1 0.0 7.619853e-24
nsim x y Sig sim\_result Exact\_result
1 10000 -1 3.5 0.1 0.0 7.619853e-24
nsim x y Sig sim\_result Exact\_result
1 100000 -1 3.5 0.1 0.0 7.619853e-24
nsim x y Sig sim\_result Exact\_result
1 10000 1 0.5 0.1 1.0 0.999999
nsim x y Sig sim\_result Exact\_result
1 100000 1 0.5 0.1 1.0 0.999999
nsim x y Sig sim\_result Exact\_result
1 10000 1 1.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 100000 1 1.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 10000 1 2.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 100000 1 2.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 10000 1 3.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 100000 1 3.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 10000 2 0.5 0.1 1.0 0.999999
nsim x y Sig sim\_result Exact\_result
1 100000 2 0.5 0.1 1.0 0.999999
nsim x y Sig sim\_result Exact\_result
1 10000 2 1.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 100000 2 1.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 10000 2 2.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 100000 2 2.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 10000 2 3.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 100000 2 3.5 0.1 1.0 1.0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 0.5 1.5 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -2 0.5 1.5 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 1.5 1.5 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -2 1.5 1.5 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 2.5 1.5 0.0485 0.043421
nsim x y Sig sim\_result Exact\_result
1 100000 -2 2.5 1.5 0.04401 0.043421
nsim x y Sig sim\_result Exact\_result
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nsim x y Sig sim\_result Exact\_result
1 100000 -2 3.5 1.5 0.08112 0.081396
nsim x y Sig sim\_result Exact\_result
1 10000 -1 0.5 1.5 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -1 0.5 1.5 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -1 1.5 1.5 0.0951 0.093837
nsim x y Sig sim\_result Exact\_result
1 100000 -1 1.5 1.5 0.09471 0.093837
nsim x y Sig sim\_result Exact\_result
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nsim x y Sig sim\_result Exact\_result
1 100000 -1 2.5 1.5 0.20657 0.204702
nsim x y Sig sim\_result Exact\_result
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nsim x y Sig sim\_result Exact\_result
1 100000 -1 3.5 1.5 0.24368 0.242677
nsim x y Sig sim\_result Exact\_result
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nsim x y Sig sim\_result Exact\_result
1 100000 1 0.5 1.5 0.26085 0.261117
nsim x y Sig sim\_result Exact\_result
1 10000 1 1.5 1.5 0.5877 0.588852
nsim x y Sig sim\_result Exact\_result
1 100000 1 1.5 1.5 0.58863 0.588852
nsim x y Sig sim\_result Exact\_result
1 10000 1 2.5 1.5 0.7004 0.699717
nsim x y Sig sim\_result Exact\_result
1 100000 1 2.5 1.5 0.70049 0.699717
nsim x y Sig sim\_result Exact\_result
1 10000 1 3.5 1.5 0.7355 0.737692
nsim x y Sig sim\_result Exact\_result
1 100000 1 3.5 1.5 0.7376 0.737692
nsim x y Sig sim\_result Exact\_result
1 10000 2 0.5 1.5 0.2626 0.261117
nsim x y Sig sim\_result Exact\_result
1 100000 2 0.5 1.5 0.26085 0.261117
nsim x y Sig sim\_result Exact\_result
1 10000 2 1.5 1.5 0.6843 0.682689
nsim x y Sig sim\_result Exact\_result
1 100000 2 1.5 1.5 0.68124 0.682689
nsim x y Sig sim\_result Exact\_result
1 10000 2 2.5 1.5 0.8641 0.860998
nsim x y Sig sim\_result Exact\_result
1 100000 2 2.5 1.5 0.86187 0.860998
nsim x y Sig sim\_result Exact\_result
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1 100000 2 3.5 1.5 0.89898 0.898973
nsim x y Sig sim\_result Exact\_result
1 10000 -2 0.5 4 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -2 0.5 4 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 1.5 4 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -2 1.5 4 0.0 0
nsim x y Sig sim\_result Exact\_result
1 10000 -2 2.5 4 0.0426 0.042552
nsim x y Sig sim\_result Exact\_result
1 100000 -2 2.5 4 0.04187 0.042552
nsim x y Sig sim\_result Exact\_result
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nsim x y Sig sim\_result Exact\_result
1 100000 -2 3.5 4 0.11763 0.117751
nsim x y Sig sim\_result Exact\_result
1 10000 -1 0.5 4 0.0 0
nsim x y Sig sim\_result Exact\_result
1 100000 -1 0.5 4 0.0 0
nsim x y Sig sim\_result Exact\_result
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nsim x y Sig sim\_result Exact\_result
1 100000 -1 1.5 4 0.04703 0.047463
nsim x y Sig sim\_result Exact\_result
1 10000 -1 2.5 4 0.1294 0.135308
nsim x y Sig sim\_result Exact\_result
1 100000 -1 2.5 4 0.13405 0.135308
nsim x y Sig sim\_result Exact\_result
1 10000 -1 3.5 4 0.2049 0.210507
nsim x y Sig sim\_result Exact\_result
1 100000 -1 3.5 4 0.20981 0.210507
nsim x y Sig sim\_result Exact\_result
1 10000 1 0.5 4 0.0973 0.099476
nsim x y Sig sim\_result Exact\_result
1 100000 1 0.5 4 0.09946 0.099476
nsim x y Sig sim\_result Exact\_result
1 10000 1 1.5 4 0.2418 0.244876
nsim x y Sig sim\_result Exact\_result
1 100000 1 1.5 4 0.24445 0.244876
nsim x y Sig sim\_result Exact\_result
1 10000 1 2.5 4 0.3275 0.332721
nsim x y Sig sim\_result Exact\_result
1 100000 1 2.5 4 0.33147 0.332721
nsim x y Sig sim\_result Exact\_result
1 10000 1 3.5 4 0.403 0.407919
nsim x y Sig sim\_result Exact\_result
1 100000 1 3.5 4 0.40723 0.407919
nsim x y Sig sim\_result Exact\_result
1 10000 2 0.5 4 0.0973 0.099476
nsim x y Sig sim\_result Exact\_result
1 100000 2 0.5 4 0.09946 0.099476
nsim x y Sig sim\_result Exact\_result
1 10000 2 1.5 4 0.2913 0.29234
nsim x y Sig sim\_result Exact\_result
1 100000 2 1.5 4 0.29223 0.29234
nsim x y Sig sim\_result Exact\_result
1 10000 2 2.5 4 0.4193 0.425477
nsim x y Sig sim\_result Exact\_result
1 100000 2 2.5 4 0.42348 0.425477
nsim x y Sig sim\_result Exact\_result
1 10000 2 3.5 4 0.4948 0.500676
nsim x y Sig sim\_result Exact\_result
1 100000 2 3.5 4 0.49924 0.500676
\end{Verbatim}
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\end{document}