4) Let \( X \) and \( Y \) be independent, standard normal random variables. Consider the polar coordinates: \[ \tan \theta=Y / X \text { and } R^{2}=X^{2}+Y^{2} \] Show that \( f_{R, \theta}(r, \theta)=\frac{1}{2 \pi} r e^{-r^{2} / 2} \)
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Step 1
First, we know that the joint density function of \(X\) and \(Y\) is given by \(f_{X,Y}(x,y) = \frac{1}{2\pi}e^{-(x^2+y^2)/2}\) since \(X\) and \(Y\) are independent standard normal random variables. Show more…
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