1 Uniform Distribution
The simplest discrete distribution where every outcome has the same probability.
Definition
$$ p(x) = \begin{cases} \frac{1}{n} & x \in \{1, 2, \dots, n\} \\ 0 & \text{otherwise} \end{cases} $$
Interactive: Fair Die Roll (n=6)
2 Binomial Distribution
Models the number of successes in \(n\) independent Bernoulli trials.
Definition
\(X \sim \text{Binomial}(n, p)\) with PMF: $$ P_X(k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k=0, 1, \dots, n $$
Interactive: Binomial Simulator
3 Poisson Distribution
Models the number of events occurring in a fixed interval with constant average rate \(\lambda\).
Definition
\(X \sim \text{Poisson}(\lambda)\) with PMF: $$ P_X(k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k=0, 1, 2, \dots $$
Interactive: Poisson Rate Adjuster
Binomial → Poisson Limit
As \(n \to \infty\) and \(p \to 0\) such that \(np = \lambda\) remains constant, the Binomial distribution converges to a Poisson distribution.