What is a nonarithmetic distribution? can give an example?

Question

Please, I need some help understanding what is a non-arithmetic distribution.

I found this definition for Arithmetic Distribution here:

A discrete probability distribution concentrated on a set of points of the form $\pm nh$, where $h>0$ and $n=1,2,...$.

But don't really understand it.

To be clear, is a Bernoulli distribution non-arithmetic? It seems it is not.

Thanks!

Answer 1

Your quoted definition is apparently for an arithmetic distribution and I would be surprised if the value $0$ was not allowed.

In effect, given a random variable $X$, if there is any positive real number $h$ such that $Y=\frac1h X$ is a random variable which can only take integer values, then $X$ has an arithmetic distribution.

If there is no such $h$ then $X$ has a non-arithmetic distribution. Examples include

• any continuous distribution
• $P(X=\frac 1n) = \frac6{\pi^2 n^2}$ for positive integer $n$