# Intuition behind moments of random variables [duplicate]

I am looking to understand the intuition behind moments of random variables.

I understand the first moment relating to the mean and the second one relating to the variance. But what use does the n-th moment have?

Thank you

## marked as duplicate by Hans Lundmark, Community♦May 22 '18 at 14:58

$$M_X(t) = \mathbb{E}\left[e^{tX}\right] = \mathbb{E}\left[\sum_k \frac{t^k}{k!}X^k\right] = \sum_k \frac{t^k}{k!}\mathbb{E}[X^k] = \sum_k \frac{t^k}{k!}\left.\frac{{\rm d}^k M(t)}{{\rm d}t^k}\right|_{t=0}$$
If you know all moments $\mathbb{E}[X^k]$ (or equivalently $M_X(t)$) you could try to invert the problem and actually estimate the underlaying PDF, this is know as the moment problem