Clément Canonne @ccanonne.github.io · Apr 28 The first "obvious" approach is to get to the definition of expectation, write the sum or integral, try to compute it, probably suffer for a while, and either eventually succeed or choose to go raise sheep in the middle of France. For illustration, let's consider X~Poisson(λ). What is 𝔼[X⁴]?
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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).