A description of the mean of the Geometric Distribution - is it unorthodox or just incorrect? 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?

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