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I've noticed that some undergraduate introductory probability textbooks and courses emphasize or seem to prefer the moment generating function $m(t) = \mathbf E(e^{tX})$ of a random variable $X$ rather than the characteristic function $f(t) = \mathbf E(e^{i t X})$. However, the characteristic function seems to have nicer properties.

Is there a reason for an undergraduate introduction to probability course to emphasize the moment generating function instead of the characteristic function? Or is the characteristic function simply better. I wonder if the characteristic function is avoided only because some students feel uncomfortable with complex numbers.

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  • $\begingroup$ I would think an intro course would cover combinatrics, pdfs, hypothesis tests (p values, t tests, normal distributions, and Bayes thereom at the level of basic dependencies. These functions you are talking about sound more advanced and theoretical. $\endgroup$
    – guest
    Aug 3, 2018 at 19:57
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    $\begingroup$ Aww man, another place my education was dumbed down for the misplaced fear of complex numbers. I remember moment generating functions, but we said nothing I recall about this characteristic function. It was a junior level course with lots of math prereqs as well. $\endgroup$
    – James S. Cook
    Aug 3, 2018 at 22:03

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At least around here, complex calculus just isn't part of the regular engineering curricula. To introduce generating functions already makes most brains explode...

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