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:

M_{X} (t) = E [\exp (t X)] .

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

M_{X}^{'} (0) = E [X], M_{X}^{″} (0) = E [X^{2}] .

Essential Result

$Var (X) = M_{X}^{″} (0) - [M_{X}^{'} (0)]^{2}$ .

Small Example

For Bernoulli $X$ , $M_{X} (t) = 1 - p + p e^{t}$ . Then $M_{X}^{'} (0) = p$ , $M_{X}^{″} (0) = p$ , and $Var (X) = p (1 - p)$ .

Common Mistakes

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

Connections

References

MIT OpenCourseWare, "6.041SC Probabilistic Systems Analysis and Applied Probability", Fall 2013, https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ ↩︎