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Recently I have seen random variable distributions described in two ways:

$$ X \sim Nb(r,p) \\ X \stackrel{d}{=} Nb(r,p) $$

Both indicating that $X$ is a negative binomial random variable with $r$ successes and $p$ probability of failure. Do these mean the same things? Are there any other ways of writing this?

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Yes.

Both symbols are commonly used to mean "has a distribution of".

I've encountered "$\sim$" much more often than "$\mathop{=}^d$", possibly because it is more convenient to typeset, but your mileage may vary.

Negative binomial is typically indicated by both-capitalised "NB", or the whole word "Pascal".

I cannot recall any other variations, but it is quite possible they exist.  A good text book should have a glossary of symbols used.

$$X\sim\mathrm{N\!B}(r,p)$$

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