First, apologies if this is answered elsewhere and I've just not managed to search it properly.
If I have a discrete random variable $X$ that follows the binomial distribution, then what distribution does $\frac{1}{X}$ have? In particular, I'd like to understand the expected value and variance. I've tried calculating these by hand and have gotten stuck.
Is this a well-studied distribution? Is there a special name for it, or something I could search to learn more about it? Someone suggested to me that it's the negative (aka inverse) binomial distribution, but I can't get the details to work out, so I think it's something else.
Thanks!
EDIT: As Henry pointed out, a binomial distribution has a nontrivial chance of being 0. For my particular application of this, it's very natural to omit these from the domain, but I suppose then it's not a strict binomial distribution.