# What's the name of the ratio between successive moment of random variable?

Let $$X$$ be a random variable.

Is there a name to the ratio: $$\frac{E[X^{n+1}]}{E[X^{n}]}$$, where $$E[\cdot]$$ is the expectation of $$\cdot$$?

We let $$n\ge0$$ and for $$n=0$$ it's simply the expectation of the constant number $$1$$.

Specifically, what is the name of this ratio when $$n=1$$: $$\frac{E[X^{2}]}{E[X]}$$?

Is there a meaningful interpenetration for this value?

• This ratio often does not exist (consider any centered random variable for instance). At a minimum, then, if anyone had ever chosen to name something like it, they likely would have studied $E[|X|^{n+1}] / E[|X|^n]$ instead. In the case $n=2,$ your ratio has a simple algebraic relationship with the coefficient of variation. – whuber Dec 23 '18 at 14:37
• and/or index of dispersion – gunes Dec 23 '18 at 14:44