# Probability distribution with standard deviation proportional to mean

Is there a probability distribution for which the standard deviation is proportional to the mean?

More generally, I have been searching the web for a nice table comparing the properties of probability distributions (including mean and std). Any good reference?

• Since a probability distribution's mean and SD are just numbers, we have to understand this question in the sense of a family of distributions. In fact, even that sense is too general, because to get a positive answer you could just take a basket of various distributions and pick out a few for which the SD:mean ratios are the same. I guess you are probably asking about a naturally parameterized family of distributions. – whuber Jun 27 '14 at 16:30

Is there a probability distribution for which the standard deviation is proportional to the mean?

For the standard deviation to be proportional to the mean, the mean must be able to vary, in which case we're really talking about a family of distributions. [Edit: oh, I see whuber makes the same point above.]

Such a family is sometimes said to have constant coefficient of variation.

There are many. Two common examples:

(i) The lognormal distribution with constant $\sigma$

(ii) The gamma distribution with constant shape parameter. This includes the exponential as a special case

I have been searching the web for a nice table comparing the properties of probability distributions (including mean and std).

Some books contain summary information like this.

Wikipedia doesn't have many distributions in one table, but it does have summary information on many distributions.

It's possible to fund some such tables on-line. Here is one example.

• +1. One could go further in emphasizing just how large "many" is here. Take any $p$-parameter family of distributions $\Omega=\{F_\theta\}$. Consider the function $u$ defined on $\Omega$ given by the ratio of the mean to the standard deviation: any contour (level set) of $u$ in $\Omega$ satisfies the required conditions and implicitly has $p-1$ parameters. When $p\gt 1$ this will (generically) itself be a large distributional family. For example, applying this to the Normal family gives a Normal$(|\mu|,\mu^2)$ family. In fact, this construction works even when $\Omega$ is nonparametric. – whuber Jun 27 '14 at 16:35