Mean and Variance for Random Variables X0

Question

I am looking to understand mean and variance for random variables a little better. Can someone help me with some examples or guide me in the right direction?
I am interested in seeing the differences between finite and infinite so I would like help with coming up with examples of random variables X>0 with:

Finite mean and finite variance

Finite mean but infinite variance

Infinite mean and infinite variance.

herb steinberg · Answer

Simple examples:  density function $f_n(x)=frac{c_n}{1+x^n}$ for $xge 0$ and $f_n(x)=0$ otherwise.  For $n=2$ mean and variance are infinite, for $n=3$ finite mean and infinite variance, for $n=4$ mean and variance finite.
$c_n$ defined so that $int_0^infty f_n(x)dx=1$.

Mean and Variance for Random Variables X>0

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