Solved – Converting central moments to non-central moments (and back)

The central moments of a probability distribution $p(x)$ are defined as:

$$theta_n = langle (x – langle x rangle)^n rangle $$

while the non-central moments are the standard:

$$mu_n = langle x^n rangle $$

By the binomial theorem, we have:

$$theta_n = sum_{k=0}^n binom{n}{k}(-1)^{n-k} mu_k mu_1^{n-k}$$

which allows us to compute the central moments from the non-central moments. Is there an inverse to this expression, giving the non-central moments $mu_n$ from the central moments $theta_n$?

One can write:

$$mu_n = langle (x – langle x rangle + langle x rangle)^n rangle$$

By the binomial theorem

$$mu_n = sum_{k=0}^n binom{n}{k} theta_k mu^{n-k}_1$$

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