Complexity of generating non-uniform random variates
The binomial, negative binomial, and hypergeometric distributions are discrete distributions. Knuth and Yao (1976) gave complexity results for discrete distributions in general. Given a stream of i.i.d. unbiased random bits, any algorithm that samples from a discrete distribution will require at least—$$b = -\sum_{x\in\Omega} \mathbb{P}(X=x)*\log_2(\mathbb{P}(X=x))$$random bits on average to sample a variate $X$ from that distribution (where $\Omega$ is the distribution's support), and they gave an algorithm that comes within 2 bits of this lower bound (and showed that coming within 2 bits is the best possible for any algorithm).
The case of continuous distributions such as the normal distribution is more complicated, but Devroye and Gravel gave complexity results for sampling a continuous variate to a desired accuracy, given a stream of i.i.d. unbiased random bits, using Wasserstein distance as the accuracy metric.
I also want to note one more thing. If the distribution of $X$ has a continuous and bounded quantile function (inverse cumulative distribution function) $F^{-1}$, I suspect that the bit complexity of sampling $X$ also depends on the modulus of continuity of $F^{-1}$, call it $\omega(h)$. In this sense, Lipschitz continuous quantile functions (where $\omega(h) = O(h)$) are the simplest cases of such functions and in that case it's simple to describe an algorithm to sample $X$ to a desired accuracy. For such functions, the algorithm uses—$$ceil(\ln(\max(1, L))-\ln(\epsilon))/\ln(\beta))$$random digits, where $L$ is the Lipschitz constant, $\epsilon$ is the desired accuracy, and $\beta$ is the digit base (such as 2 for binary). I suspect that with other moduli of continuity, the complexity will be as follows: $$ceil(-\ln(\omega^{-1}(\epsilon))/\ln(\beta)).$$
REFERENCES:

Knuth, Donald E. And Andrew Chi-Chih Yao. &quot;The complexity of nonuniform random number generation&quot;, in Algorithms and Complexity: New Directions and Recent Results, 1976.
Devroye, L., Gravel, C., &quot;Random variate generation using only finitely many unbiased, independently and identically distributed random bits&quot;, arXiv:1502.02539v6 [cs.IT], 2020.