Let's say a friend tells me he needs my help for chucking wood. He tells me that this takes 10 minutes on average. This motivates my following question.
Given an expectation value E on a positive random variable and assuming that it's variance exists, is there a limit on the variance of this variable or can it be arbitrary large?
The variance may even be infinite. If we want finite very large variance, let $X$ for example have density functions of the form $$\frac{k\alpha}{(1+kx)^{\alpha+1}}$$ for $x\ge 0$.
Here $k$ and $\alpha$ are positive parameters. For the variance of $X$ to exist, we need $\alpha\gt 3$.
Given a target positive mean $\mu$, and $\alpha\gt 2$, we can find a $k=k(\alpha,\mu)$ such that the mean of $X$ is $\mu$. And by taking $\alpha=3+\epsilon$, where $\epsilon$ is small enough positive, we can then make the variance of $X$ arbitrarily large, while the expectation of $X$ stays fixed at $\mu$.
