Clément Canonne @ccanonne.github.io · Apr 28 The above computation is possible! It can be done. It's just not pleasant. And then, if after that you want to compute 𝔼[X⁵] as well... Another approach: the Moment-Generating Function (MGF)! It does not always exist, but when it does, it encodes all moments a random variable X (hence the name).
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Now, you can do a lot of things with MGFs, but a nice one? Differentiating them. Namely, it's a simple exercise to check that the n-th derivative of M_X(t), evaluated at t=0, is exactly the n-th moment 𝔼[Xⁿ] of X! Modulo a couple missing details, the proof is below. Great! But... so what?