discretely distributed

Source

Made it through chapter 5 of Hassett and Stewart, which details six discrete probability distributions whose properties I would like to be able to commit to memory. For each, where $X=k$ is the random variable where $k$ events are drawn from the distribution, the probability mass function, expectation value, and variance are as below.

distribution $P(k)$ $E(X)$ $V(X)$
binomial $${n\choose k}p^k q^{n-k}$$ $$np$$ $$npq$$
hypergeometric $$\frac{{K\choose k}{N-K\choose n-k}}{N\choose n}$$ $$\frac{nK}{N}$$ $$\left(\frac {nK}{N}\middle) \middle(1-\frac KN\middle) \middle(\frac{N-n}{N-1}\right)$$
Poisson $$\frac{e^{-\lambda} \lambda^k} { k!}$$ $$\lambda$$ $$\lambda$$
geometric $$q^k p$$ $$\frac qp$$ $$\frac q{p^2}$$
negative binomial $${r+k-1 \choose r-1}q^k p^r$$ $$\frac {nq}p$$ $$\frac{nq}{p^2}$$

In each of these cases, $P(S) = p$ is the probability of single success, $q=1-p$ is the probability of a single failure, and the remaining parameters are as follows:

  • binomial
    Sampling from a total of $n$ independent trials, the probability of $k$ successes.
  • hypergeometric
    Sampling "without replacement" from a finite population $N$, of whom $K$ individuals are "successes." Here $P(k)$ is the probability of $k$ successes drawn from the subpopulation, out of $n$ total samples.
  • Poisson
    Here $P(k)$ gives the probability of $k$ completely independent events, in a scenario where the mean number of events over many such scenarios is $\lambda$.
  • geometric
    A series of independent experiments is repeated until the first success; $P(k)$ is the probability that there are $k$ initial failures.

    Here it's useful to know the cumulative distribution function $$F(n) = \sum_{k<n} P(k) = 1-q^n$$ as well. There are lots of opportunities for fencepost errors.

  • negative binomial
    Extending the geometric distribution to the question of whether the $r$-th event is preceded by exactly $k$ others.

I don't know whether it will be useful to know other cumulative distributions.1

  1. Note added [2025-07-02 Wed]: The Poisson CDF sucks. ↩