Generating Functions, Taylor Series, Taylor Theorem, Central Limit Theorem, Deep Dives.
0. We’ve seen that these derivatives of the MGF happen to be the 0-th, 1st, 2nd, etc. moments of Z. The 0-th moment, M⁰(0), is always 1. Recall that Z is, by its construction, a standard normal random variable. Hence, its first moment (mean), M¹(0), is 0, and its second moment (variance), M²(0), is