Question

(A derivation of the bivariate normal distribution) let $Z_{1}$ and $Z_{2}$ be independent n(0,1) random variables, and define new random variables X and Y by\\
\begin{align*}
X=a_{x}Z_{1}+b_{x}Z_{2}+c_{x} \quad Y=a_{Y}Z_{1}+b_{Y}Z_{2}+c_{Y}\\
\end{align*}
where $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ are constants.\\
if we define the constants $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ by\\
\begin{align*}
a_{x}=\sqrt{\frac{1+\rho}{2}}\sigma_{X}, b_{x}=\sqrt{\frac{1-\rho}{2}}\sigma_{X}, c_{X}=\mu_{X},\\
a_{Y}=\sqrt{\frac{1+\rho}{2}}\sigma_{Y}, b_{Y}=-\sqrt{\frac{1-\rho}{2}}\sigma_{Y},c_{Y}=\mu_{Y}\\
\end{align*}
where \mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho are constants\\
\\
\begin{itemize}
\item
Question 1):\\
Show that $(X,Y)$ has the bivariate normal pdf with parameter\\s $\mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho$\\
\item
Question 2):\\
if we start with bivariate normal parameters $$\mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho$, define constants $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ as the solutions to the equations:\\
\begin{align*}
\mu_{X}=c_{X}, \sigma^2_{X}=a^2_{X}+b^2_{X},\\
\mu_{Y}=c_{Y}, \sigma^2_{Y}=a^2_{Y}+b^2_{Y},\\
\rho\sigma_{X}\sigma_{Y}=a_{X}a_{Y}+b_{X}b_{Y}
\end{align*}
  
\end{itemize}

\end{document}

Answer 1

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