Moment generating function is the expected value of $e^{tx}$ with respect to the pdf of the variable. If we differentiate mgf and then let $t=0$, we can get $E(X), E(X^2), \dots$. So mgf is a way to find $\mu$ and $\sigma$ if the pdf itself is complicated. Furthermore, mgf can also be used to prove the similarity of two pdfs, if and only if their mgf are the same.
Moment Generating Function
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