Exponential, Gamma, Beta

Inter-arrival times, sums of exponentials, and modeling probabilities themselves.

Three continuous distributions that come up everywhere in modeling.

The Exponential $(\lambda)$ distribution has density $f(x) = \lambda e^{-\lambda x}$ for $x \ge 0$ , with mean $1/\lambda$ . It's the continuous analog of the Geometric and is memoryless. It models the inter-arrival time of a Poisson process — wait times between trades, between defaults, between bus arrivals.

The Gamma $(k, \theta)$ distribution has density proportional to $x^{k-1} e^{-x/\theta}$ , generalizing the Exponential. The sum of $k$ independent Exponentials is Gamma, so it models the time until the $k$ -th arrival.

The Beta $(\alpha, \beta)$ distribution lives on $[0, 1]$ :

f(x) \propto x^{\alpha - 1}(1 - x)^{\beta - 1}

It's the natural distribution for an unknown probability. Beta is conjugate to the Bernoulli/Binomial: a Beta prior plus binomial data gives a Beta posterior, and the update is just adding observed successes to $\alpha$ and failures to $\beta$ . That makes Bayesian updating mechanical for click-through rates, win rates, and any other probability you're trying to estimate.