We recommend reading this documentation on
https://alexpghayes.github.io/distributions3/, where the math
will render with additional detail.
In the following, let \(X\) be a Poisson random variable with parameter
lambda = \(\lambda\).
Support: \(\{0, 1, 2, 3, ...\}\)
Mean: \(\lambda\)
Variance: \(\lambda\)
Probability mass function (p.m.f):
$$
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
$$
Cumulative distribution function (c.d.f):
$$
P(X \le k) = e^{-\lambda}
\sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}
$$
Moment generating function (m.g.f):
$$
E(e^{tX}) = e^{\lambda (e^t - 1)}
$$