Clément Canonne @ccanonne.github.io · Apr 28 But wait, I still need to differentiate that, right? I want the n-th derivative of this thing evaluated at 0. That doesn't seem "simple" or "enjoyable." Yeah, it's not. Thankfully, you don't need to! You don't actually want the n-th derivative, you see. You want it *at 0.* Enter Taylor expansions.
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Let's look again at X~Poisson(λ). We have its MGF: let's Taylor expand it at 0, to order 4. (With some practice, it's easy and systematic, and anyways, Wolfram Alpha exists.) That's all, only need to read the coef. 𝔼[X⁴]=λ(λ³+6λ²+7λ+1). Want 𝔼[X³], 𝔼[X⁵] too? No problem, Taylor expand a bit more!