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I have a homework assignment where I'm asked to propose an estimator for the mean of a geometric random variable. This seemed simple enough, given that I've always understood the mean of the geometric distribution to be $1\over{p}$. The directions, however, describe it this way:

Recall that the mean of a geometric random variable $X$ is given by $\theta = \exp(\mathbb E[\log(X)])$

I cannot understand this construal of the mean of a geometric distribution. I've tried manipulation with MGFs and it still doesn't make sense. Is there a typo in the homework directions or is this legitimately a way of describing the mean of X? If it's the latter, how does it work?

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$\exp(\mathbb E[\log(X)])$

is the geometric mean of a positive random variable $X$

not the mean of a geometric random variable.

So either the homework directions put the words in the wrong order, or you transcribed them incorrectly

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    $\begingroup$ Seems like the directions put the words in the wrong order. Thank you! $\endgroup$
    – jmars
    Commented Dec 9, 2021 at 23:01

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