The moment generating function (MGF) of a random variable, M(t) = E(e^(tX)), encodes all its moments: when M(t) has a valid Taylor/power series expansion, the coefficient of t^n/n! equals the nth moment E(X^n), allowing moments to be obtained via pattern recognition or differentiation rather than direct integration. MGFs also determine a distribution uniquely (two random variables with the same MGF have the same distribution) and convert sums of independent random variables into products of MGFs, which is exploited to prove that sums of independent Poisson random variables are themselves Poisson. This belongs to probability theory, specifically the study of moments and transform methods as an alternative to direct computation via LOTUS (law of the unconscious statistician).
The moment generating function (MGF) of a random variable, M(t) = E(e^(tX)), encodes all its moments: when M(t) has a valid Taylor/power series expansion, the coefficient of t^n/n! equals the nth mom…