Generating Functions, Taylor Series, Taylor Theorem, Central Limit Theorem, Deep Dives.
above equation, set x = t²/2 and substitute this result in the R.H.S. of Eq. (15), and you have proved the Central Limit Theorem: As the sample size tends to infinity, the MGF of the standardized sample mean equates to the MGF of the standard normal random variable (Image by Author) (Image by