Why the moment-generating function $M(t)=E\left(e^{tx} \right)$ does not always exist while the characteristic function $\varphi(\epsilon)=E\left(e^{i\epsilon x}\right)$ exists for any random variable X?
2026-03-31 07:26:13.1774941973
Why the moment-generating function does not always exist
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Counter example is Cauchy distribution.
moment generating funciton does not exist , but characteristic function exists.
Since $E(e^{tx})$ is not absolutely convergent