Let x and y be independent standard normal random variables

Question

Let X and Y be independent standard normal random variables. Find the mgf of X²+Y². What can you conclude about the distribution of X²+Y ²? (Hint: See Example 9.19.)

Example 9.19

Sum of i.i.d. exponentials is gamma. We have shown that the sum of i.i.d. exponential random variables has a gamma distribution. Here we prove the result using the mgfs. First find the mgf of the exponential distribution.