<!--
.. title: discretely distributed
.. slug: discretely-distributed
.. date: 2025-06-30 09:16:35 UTC-05:00
.. tags: probability, study
.. category: actuary
.. description: In which I try to recall the discrete distributions.
.. type: text
.. has_math: true
-->

Made it through [chapter 5 of Hassett and
Stewart](/pages/study/hassett/chapter-5), 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.

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| 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](/posts/2025/p-exam/p-07-02).