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Most probability books start by talking about four very different things: mean, variance, skewness and kurtosis and then miraculously there is a common thread across all these - these are all so-called "moments" of a distribution. To make matters even more amazing, there is a mysterious function - the moment-generating function/mgf, which allows one to find the rth moment of a distribution by simply taking the rth derivative of the mgf and evaluating it at zero. Magic! The student can imagine a world back in time when the concepts of mean, variance, skewness, and kurtosis existed and then some big prize was announced for someone who would discover a function that would produce all these and more (higher moments). Whoever then would have discovered the mgf should be a household name today as this discovery would seem like nothing short of miraculous; a discovery up there with that of penicillin, perhaps.

How did early probability pioneers stumble upon the concept of mgf? Why would it be integral of an exponential function? Kind of random, isn't it? Why would anyone think that such a function would even exist, which would miraculously give the four unrelated things we just learned about (mean, variance, skewness, and kurtosis) and more? All seems too good to be true, too glorious and incredible of a discovery. Or did people discover the moment generating function first and decided to name the first 4 of its moments - the mean, variance, skewness, and kurtosis? Shouldn't the Moment Generating Function be taught first and then the mean, variance, skewness, and kurtosis after that as just alternative names to the first four moments, to parallel the likely order of discovery? As things are currently taught I imagine some students telling themselves "How in the world could someone discover the mgf to magically produce the four unrelated things I just learned (mean, variance, skewness, and kurtosis)? How would anyone even think such a thing exists, which magically gives the mean, variance, skewness, and kurtosis? Hopefully the student doesn't follow up by saying that "Perhaps this field is not for me as I can never see myself making such a discovery (which, I suspect, in reality was a never much of a discovery to begin with)".

ColorStatistics
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    While the writing style is a little...discursive, there is a fairly cogent question in there: how were moment-generating functions discovered? (or possibly, how could I have discovered a mgf?

    I don't know the answer, but this thread might help: https://hsm.stackexchange.com/questions/3420/what-is-the-history-of-moment-generating-functions-and-the-more-general-charact

     – Matt Krause Feb 17 '20 at 15:49
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    Perfect reference. Thank you. – ColorStatistics Feb 17 '20 at 16:24
  • One learns about mean, standard deviation/variance, and skewness long before one has the tools or motive to learn about mgf and higher moments. One cannot say "Don't compute an average until you learn about mgf" any more than one can say "Don't turn on the lights until you understand electronics." That is, most people never learn either about mgf or how artificial light is made, and even fewer know both. – Carl Apr 07 '20 at 15:47
  • That is a great point, Carl. I am in agreement with you there. – ColorStatistics Apr 07 '20 at 17:40
  • Some advice. As things stand now, opinions are divided as to whether this question is more of a rant or an actual question. If you were to rephrase and posit this more to the effect of asking for analogies from mathematics to which mfg shows structural similarity it will likely be reopened. – Carl Apr 07 '20 at 19:46
  • I think that from inception this post was not going to get much love on this forum given the audience. Statisticians who have learned about the M.G.F. decades ago would probably be more likely to find this post to be a rant. Younger practitioners, especially those who entered data analysis/statistics from the econometrics side are more likely to relate to the point I am trying to make here. I do not particularly care whether the question is reopened or not. – ColorStatistics Apr 07 '20 at 20:19

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