Intuitive explanation of variance and moment in Probability While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. 
What is a good way to think of those two terms?

The localization and the dispersion probabilistic measures can be "seen" as the corresponding momentums of mechanical systems of "probabilistic masses".
The expectation has the following mechanical interpretation. Given that $F(x)$ is the "probabilistic mass" contained in the interval $0\le X\lt x$ (in one dimention), the mathematical expectation of the random variable $X$ is the static momentum with respect to the origin of the "probabilistic masses" system.
The variance is the mechanical analog of the inertia momentum of the "probabilistic masses" system with respect to the center of masses.
The variance of the random variable $X$ is the 2nd order momentum of $X-m$ where $m$ is the expectation of $X$, i.e. $m=E(X)=\displaystyle\int_0^{x} x\, dF(x)$ ($F(x)$ is the cumulative distribution function).
The $k$-order momentum is the expectation of $(X-m)^k$.
