A family of closely related discrete distributions arises from independent Bernoulli trials. The Bernoulli distribution itself models a single binary outcome: \(P(X = 1) = p\), \(P(X = 0) = 1 - p\), with mean \(p\) and variance \(p(1 - p)\). The Binomial distribution counts the number of successes across \(n\) independent Bernoulli(\(p\)) trials, with PMF \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) for \(k = 0, 1, \dots, n\), mean \(np\), and variance \(np(1-p)\). Generalizing from two outcomes to \(K\) categories gives the Categorical (or Multinoulli) distribution, which samples a single category with probabilities summing to one, and the Multinomial distribution, which counts outcomes across \(K\) categories in \(n\) independent draws with joint PMF \(\frac{n!}{\prod_i x_i!} \prod_i p_i^{x_i}\).
Several other discrete distributions handle waiting-time and event-counting problems. The Geometric distribution (in its "number-of-trials" form) gives the trial index of the first success, \(P(X = k) = (1-p)^{k-1} p\) for \(k = 1, 2, \dots\), with mean \(1/p\); the equivalent "failures-before-success" form is \(P(X = k) = (1-p)^k p\) starting at \(k = 0\), with mean \((1-p)/p\). Extending this idea, the Negative Binomial distribution counts failures before the \(r\)-th success, with PMF \(P(X = k) = \binom{r+k-1}{k} p^r (1-p)^k\), mean \(r(1-p)/p\), and variance \(r(1-p)/p^2\). The Poisson distribution models counts of independent events at constant rate \(\lambda\) over a fixed interval, \(P(X = k) = e^{-\lambda} \lambda^k / k!\), with both mean and variance equal to \(\lambda\). It is a good model for rare events such as decay events, web page hits per minute, or arrival counts.
The Poisson distribution emerges naturally as a limit of the Binomial: when \(p \to 0\) and \(n \to \infty\) with \(np = \lambda\) held fixed, \(P(X = k) \approx e^{-\lambda} \lambda^k / k!\). This justifies using Poisson as the "rare events" approximation. As a useful implementation detail, \(k\) independent Bernoulli(0.5) samples can be combined into a uniformly distributed integer in \(\{0, \dots, 2^k - 1\}\) by interpreting the bit string as binary — an example of how simple distributions compose to produce more complex ones.