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We are told that if $X \sim N(0,1)$, then $X^2$ has gamma distribution. Also, if $Y \sim N(0,1)$ and is independent from $X$, then $X^2 + Y^2$ has gammma/chi-squared distribution.

Let $(X,Y)$ be a random point whose joint distribution is $f(x,y)=(1/2\pi)e^{-x^2/2}e^{-y^2/2}$ for real numbers $x$ and $y$ which are independent standard normal random variables. Then $R=\sqrt{X^2+Y^2}$ is their distance to the origin. Find the distribution of $R$.

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Hint: For $a \geq 0$, $P\{R \leq a\}$ is the probability that the random point $(X,Y)$ lies in the disk of radius $a$ centered at the origin. Do you know how to calculate such probabilities from the joint density function of $(X,Y)$? If so, do the computation and you will have found the CDF $F_R(a)$. –  Dilip Sarwate Nov 1 '12 at 23:21
Any luck with the road-to-success @Dilip suggested? –  Did Nov 19 '12 at 7:59
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