TITLE: What is the use of moments in statistics?
QUESTION [36 upvotes]: Can anyone give me a simple explanation (i.e. without too many equations) of what is the use of moments in statistics? Why do we need moments? What can we learn from them?

REPLY [4 votes]: A possible intuition is as follows. Suppose that we have a random variable $X$ and let us consider the probability $P(|X|>x)$ with some $x>0$. We know that $P(|X|>x)\to0$ as $x\to\infty$ (we can prove this using the properties of the cumulative distribution function). But we might be interested in the rate of convergence. For example, can we say that $$x^pP(|X|>x)\to0$$ as $x\to\infty$ with some $p>0$? If $\operatorname E|X|^p<\infty$, then this is actually the case. Also, if $x^pP(|X|>x)\to0$, then $\operatorname E|X|^q<\infty$ for each $q<p$. So moments establish a certain bound on the rate of convergence of $P(|X|>x)$ and this rate of convergence is important in many situations (for instance, the law of large numbers and the central limit theorem). Loosely speaking, the more moments we have, the less probable large values of the random variable are.