In a Poisson distribution the square of the standard deviation $\sigma$ is equal to mean $\mu$ ($\sigma^2=\mu$) and in a binomial distribution $\sigma ^2=\mu\,(1-p)$ (with $p$ the probability of success).
I wonder what relations exist between the mean and the standard deviation in other random processes.
Does the standard deviation always increase with the mean?
Are they always related or may be independent?
Particular cases are also welcome.