If $Z_1,\ldots, Z_n$ are independent standard normal random variables, then the random variable $X = \sum_i Z_i^2$ is said to have a chi-square distribution with $n$ degrees of freedom.
If you compute the moment generating function of $X$, you find (after a fair amount of calculation) that it is the moment generating function of a gamma random variable with parameters $(n/2,1/2)$. This shows that the chi-square distribution with $n$ degrees of freedom is the gamma distribution with parameters $n/2$ and $1/2$.
However, this result seems like a complete surprise to me. I did not see that coming. Is there any reason to expect this result to be the case, or any way to more directly or intuitively understand this connection between chi-square and gamma random variables?