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.

  • 2
    $\begingroup$ In a one-parameter family, mean and variance will of necessity be related. In a two parameter family (most common example: the normal) they are not necessarily related. $\endgroup$ – André Nicolas Nov 2 '13 at 1:46
  • $\begingroup$ @AndréNicolas Thanks. Sure in the one-parametric case, like Poisson, obvious. In the binomial they are also related: the variance increases with the mean. Under which conditions are they related in the multi-parameter case? And when there is a single parameter, is the variance always an increasing function of the mean? $\endgroup$ – drake Nov 2 '13 at 1:56
  • $\begingroup$ In any family in which if $X$ has distribution in the family, $X+k$ also is, the variance is unrelated to the mean. Beside normal, a natural example is the uniforms on $[a,b]$. For one-parameter families, I suspect there would in general be increase of variance as the mean increases. One can go down the Wikipedia list of named probability distributions. $\endgroup$ – André Nicolas Nov 2 '13 at 2:10

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