It would be appreciated if the following examples could be given:


A distribution with infinite mean and infinite variance.
A distribution with infinite mean and finite variance.
A distribution with finite mean and infinite variance. 
A distribution with finite mean and finite variance.


It comes from me seeing these unfamiliar terms (infinite mean, infinite variance) used in an article I am reading, googling and reading a thread on the Wilmott forum/website, and not finding it a sufficiently clear explanation. I also haven't found any explanations in any of my own textbooks.

The mean and variance are defined in terms of (sufficiently general) integrals. What it means for the mean or variance to be infinite is a statement about the limiting behavior for those integrals
For example, for a continuous density the mean is $\lim_{a,b\to\infty}\int_{-a}^b x f(x)\  dx$ (which might here be considered as a Riemann integral, say).
This can happen, for example, if the tail is "heavy enough"; either the upper or the lower part (or both) may not converge to a finite value. Consider the following examples for four cases of finite/infinite mean and variance:

A distribution with infinite mean and non-finite variance.
Examples: Pareto distribution with $\alpha= 1$, a zeta(2) distribution.

A distribution with infinite mean and finite variance.
Not possible.

A distribution with finite mean and infinite variance.
Examples: $t_2$ distribution. Pareto with $\alpha=\frac{3}{2}$.

A distribution with finite mean and finite variance.
Examples: Any normal. Any uniform (indeed, any bounded variable has all moments). $t_3$.


 

These notes by Charles Geyer talk about how to compute relevant integrals in simple terms. It looks like it's dealing with Riemann integrals there, which only covers the continuous case but more general definitions of integrals will cover all the cases you will be likely to require [Lebesgue integration is the form of integration used in measure theory (which underlies probability) but the point here works just fine with more basic methods]. It also covers (Sec 2.5, p13-14) why "2." isn't possible (the mean exists if the variance exists).
