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?

  • 3
    $\begingroup$ 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. $\endgroup$ – whuber Dec 23 '18 at 14:37
  • $\begingroup$ and/or index of dispersion $\endgroup$ – gunes Dec 23 '18 at 14:44

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