# Given bivariate random variable, find the distribution [duplicate]

Let $X$ and $Y$ be two independent random variables. If $X$ and $Y$ are both standard normal, then what is the distribution of the random variable $$\frac{1}{2}(X^2+Y^2)$$

Is it right that the answer to this question is $$\frac{1}{2}(X^2+Y^2) \texttt{ ~ } EXP(1)$$? But how? Thank you very much, any help will be much appreciated.

## marked as duplicate by Did, Davide Giraudo, Namaste, Aweygan, NChMay 8 '17 at 18:39

• You can find the answer here, here, here or here – caverac May 8 '17 at 10:22
• How did it arrive to an EXP distribution? – geniwebb May 8 '17 at 11:02
• Much appreciated. Thanks! – geniwebb May 8 '17 at 12:32
• I add here the reference: Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society (2003). – user243301 May 8 '17 at 12:45