Calculating Variance and Standard Deviation Using Moment-Generating Functions

The variance of a random variable $X$ measures the spread of its distribution about the mean. We can compute it directly from the moment-generating function (MGF) using

$$\text{Var}(X) = E[X^2] - (E[X])^2 = M''_X(0) - \left[M'_X(0)\right]^2.$$

This follows because $E[X] = M'_X(0)$ and $E[X^2] = M''_X(0)$.

For example, consider the MGF

$$M_X(t) = \tfrac{1}{2}e^t + \tfrac{1}{2}e^{3t}.$$

Differentiating twice, we get

$$\begin{align*} M'_X(t) &= \tfrac{1}{2}e^t + \tfrac{3}{2}e^{3t}, & M'_X(0) &= 2, \\[3pt] M''_X(t) &= \tfrac{1}{2}e^t + \tfrac{9}{2}e^{3t}, & M''_X(0) &= 5. \end{align*}$$

Therefore,

$$\text{Var}(X) = 5 - 2^2 = 1.$$