Characteristic Function

Compact study note.

Summary

Characteristic functions use complex exponentials and uniquely determine real-valued distributions.^[1]

φ_{X} (t) = E [\exp (i t X)] .

Prerequisites

Notation and Assumptions

Use $i^{2} = - 1$ and real $t$ . Since

| \exp (i t X) | = 1,

the expectation is always finite.

Essential Result

For independent $X$ and $Y$ ,

φ_{X + Y} (t) = φ_{X} (t) φ_{Y} (t) .

If moments exist,

φ_{X}^{(n)} (0) = i^{n} E [X^{n}] .

Small Example

For $X \sim Uniform (l, u)$ ,

φ_{X} (t) = \frac{\exp (i t u) - \exp (i t l)}{i t (u - l)}, t \neq 0.

Also $φ_{X} (0) = 1$ .

Common Mistakes

Thinking characteristic functions fail to exist when MGFs fail.
Forgetting the special value at $t = 0$ in formulas with $t$ in the denominator.

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/ ↩︎