Discrete Distributions

The building blocks of probability. From coin flips to rare events, these models describe the discrete world.

1. Uniform Distribution

The simplest distribution where every outcome is equally likely. Think of rolling a fair die.

$$ P(X=k) = \frac{1}{n} \quad \text{for } k \in \{1, 2, \dots, n\} $$

2. Bernoulli Distribution

Models a single trial with two outcomes: Success (1) or Failure (0).

PMF

$$ P(X=k) = \begin{cases} p & \text{if } k=1 \\ 1-p & \text{if } k=0 \end{cases} $$
X ~ Ber(p)
Parameter: p (probability of success)

3. Binomial Distribution

The sum of \( n \) independent Bernoulli trials. It counts the number of successes in \( n \) trials.

$$ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} $$

Interactive Binomial Explorer

4. Geometric Distribution

Models the number of trials needed to get the first success.

$$ P(X=k) = (1-p)^{k-1}p \quad \text{for } k=1, 2, 3, \dots $$

Note: Some definitions count the number of failures before the first success. We use the definition counting total trials.

5. Poisson Distribution

Models the number of rare events occurring in a fixed interval of time or space. It is the limit of the Binomial distribution as \( n \to \infty \) and \( p \to 0 \) while \( np = \lambda \) remains constant.

$$ P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!} $$

Poisson Approximation of Binomial

See how the Poisson distribution (lines) approximates the Binomial distribution (bars) when n is large and p is small.

Lambda (λ) = np = 5.00