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Polar Method for Generating Normal Random Variables 12/2

The normal distribution:

\begin{displaymath}X \times Y\ \mbox{indpt.}\ {\cal N}(0, 1)
\end{displaymath}


\begin{displaymath}f_{x, y} (x, y)=\frac{1}{2\pi}-e^{-\frac{x^2}{2}} e^{-\frac{y^2}{2}}
\end{displaymath}


\begin{displaymath}R\leq 0 \quad 0\leq \sigma \leq 2\pi
\end{displaymath}


\begin{eqnarray*}R &=& \sqrt{x^2 + y^2} \\
\Theta &=& \left\{
\begin{array}{ll}...
...nd{array}\right.\\
X &=& R \cos \Theta \\
Y &=& R \sin \Theta
\end{eqnarray*}


Joint Density:

\begin{displaymath}f_{R, \Theta} (r, \theta) dr d\theta = P(r \leq R \leq r + dr...
... \theta + d \theta) =dr \times r d\theta \times \mbox{density}
\end{displaymath}


\begin{displaymath}f_{R \theta} (r, \theta) =r f_{xy} (r \cos \theta, r \sin \theta)
\end{displaymath}


\begin{eqnarray*}& = & \frac{r}{2 \pi} e^{-\left(\frac{r^2 \cos^2 \theta}{2} + \...
...\theta}{2}\right)} \\
& = & \frac{1}{2 \pi} re^{-\frac{r^2}{2}}
\end{eqnarray*}


$x=r \cos \theta$
$y=r \sin \theta$



These are independent:


\begin{displaymath}R\sim r e^{-\frac{r^2}{2}} \mbox{is Rayleigh}
\end{displaymath}


\begin{displaymath}\Theta \sim U(0, 2 \pi)
\end{displaymath}


\begin{displaymath}T=R^2 \quad f_T(t)=\frac{1}{2}e^{-\frac{t}{2}}
\end{displaymath}

because


\begin{displaymath}T=g(R)\quad g^{-1}(y)=\sqrt{y}
\end{displaymath}


\begin{displaymath}f_T(y)=f_x(\sqrt{y}) \times \vert \frac{d}{dy} \sqrt{y} \vert
\end{displaymath}


\begin{eqnarray*}T=R^2 \sim \mbox{Exp} \left( \frac{1}{2}\right) \\
\\
-2 \lo...
...\cos (2 \pi U_2) \\
\\
Y= \sqrt{-2 \log U_1} \sin (2 \pi U_2)
\end{eqnarray*}


This is known as the polar method for generating normal random variables.



Susan Holmes
1998-12-07