Is the following statement true? Every random variable has a moment generating function

No. The standard Cauchy distribution E[e^{tX}] = ∞ for every t ≠ 0; even E|X| is infinite. Its characteristic function e^{−|t|} exists.

Counterexample

Is this true?

Every random variable has a moment generating function

No — it is false.

The counterexample

The standard Cauchy distribution

Why it fails

E[ $e^{tX}] = \infty$ for every t ≠ 0; even E|X| is infinite. Its characteristic function $e^{-|t|}$ exists.

Keep practicing

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.