Expectation from a Moment Generating Function

Compact study note.

Summary

When an MGF exists near zero, derivatives at zero recover raw moments. This is a compact way to derive means and variances for many standard distributions.^[1]

Prerequisites

• Moment Generating Functions

Notation and Assumptions

MGF definition:

If differentiation under the expectation is justified near $0$ , then

Essential Result

$\mathrm{Var}(X)=M_X^{″}(0)-[M_X^{\prime }(0){]}^2$ .

Small Example

For Bernoulli $X$ , $M_X(t)=1-p+p{e}^t$ . Then $M_X^{\prime }(0)=p$ , $M_X^{″}(0)=p$ , and $\mathrm{Var}(X)=p(1-p)$ .

Common Mistakes

• Differentiating an expression outside its domain of convergence.
• Using this method for a distribution whose MGF does not exist near zero.

Connections

• Moment Generating Functions
• Expected Value
• Variance

References