# Suppose that Y_(1) and Y_(2) are independent, standard normal random variables. Find the density

Suppose that ${Y}_{1}$ and ${Y}_{2}$ are independent, standard normal random variables. Find the density
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Dwayne Small
This distribution is ${\chi }^{2}$ with v = 2. So both our random variables have $m\left(t\right)=\left(1-2t{\right)}^{-1/2},t<\frac{1}{2}$ Note that when U=A+B where A and B are independent random variables we get ${m}_{U}\left(t\right)={m}_{a}\left(t\right){m}_{B}\left(t\right)$ By this logic we get
${m}_{U}\left(t\right)=\left(1-wt{\right)}^{-1}$
Note that this is the moment generating function for an exponential random variable with $\lambda =\frac{1}{2}$. So we get the following density function.
${f}_{U}\left(u\right)=\frac{1}{2}{e}^{-u/2},u\ge 0$