Variance of Gaussian Distribution/Proof 2

Proof
By Moment Generating Function of Gaussian Distribution, the moment generating function of $X$ is given by:


 * $\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

From Variance as Expectation of Square minus Square of Expectation:


 * $\var X = \expect {X^2} - \paren {\expect X}^2$

From Moment Generating Function of Gaussian Distribution: Second Moment:


 * $\map { {M_X}''} t = \paren {\sigma^2 + \paren {\mu + \sigma^2 t}^2 } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

From Moment in terms of Moment Generating Function, we also have:


 * $\expect {X^2} = \map { {M_X}''} 0$

Setting $t = 0$, we obtain the second moment:

So: