Moment Generating Functions

Compact study note.

Summary

Moment generating functions package moments through expectations of exponentials when they exist near zero.^[1]

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

Prerequisites

Expected Value

Notation and Assumptions

Use $M_{X} (t)$ for the MGF and $K_{X} (t)$ for the cumulant generating function. Do not use nonstandard 'central MGF' or 'asymmetric MGF' like separate standard objects.

Essential Result

If $M_{X}$ exists near $0$ , then $E [X^{n}] = M_{X}^{(n)} (0)$ . If independent $X, Y$ have MGFs near $0$ , then $M_{X + Y} (t) = M_{X} (t) M_{Y} (t)$ .

Small Example

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

Common Mistakes

Assuming every distribution has an MGF near zero; log-normal does not.
Confusing an MGF with one characteristic function, which always exists.

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/ ↩︎