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Suppose that we have a random variable $X$, $i \in \mathbb N$, and a scalar $t$. Is there a proper name for these integrals, that I for the moment call 'shifted moments' ?

$$I_{i}^{t} = \mathbb{E}\left(X^i e^{tX}\right)$$

I use the name 'shifted moments' as they are derivatives of the moment generating function, but not in $t = 0$.

How would you call them ?

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  • $\begingroup$ $I_0^t$ is called the cumulant generating function; wouldn't you then just call $I_i^t$ the $i$-th derivative of this function? (up to a factor $t^i$) $\endgroup$
    – Carlo Beenakker
    Jun 11, 2020 at 9:31
  • $\begingroup$ I think $I_0^t$ is the moment generating function. It's log is the cumulant generating function. $\endgroup$
    – lrnv
    Jun 11, 2020 at 9:40
  • $\begingroup$ certainly, cumulant $\mapsto$ moment, my mistake. $\endgroup$
    – Carlo Beenakker
    Jun 11, 2020 at 9:41
  • $\begingroup$ Yes of course, these are derivatives of the moment generating function. But i watned to found a better name. If there is not, i'll take it. (Btw, i have the same issue with cumulant generating function). $\endgroup$
    – lrnv
    Jun 11, 2020 at 9:48

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These are moments of what is often called the exponentially tilted distribution.

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