Many continuous distributions arise from natural generative processes. The Uniform distribution on \([a, b]\) has constant density \(f(x) = 1/(b - a)\) over the interval, with mean \((a + b)/2\) and variance \((b - a)^2/12\). The Exponential distribution with rate \(\lambda\) models the time between events in a Poisson process; its PDF is \(f(t) = \lambda e^{-\lambda t}\) for \(t \ge 0\), with mean \(1/\lambda\) and variance \(1/\lambda^2\). A defining property of the Exponential is its memorylessness: \(P(X > s + t \mid X > s) = P(X > t)\), meaning that the distribution of remaining waiting time is independent of how long one has already waited. The Gamma distribution generalizes the Exponential via a shape parameter \(k > 0\) and rate \(\theta > 0\) (or scale \(\beta = 1/\theta\)), with PDF \(f(x) = x^{k-1} e^{-x/\theta} / (\Gamma(k) \theta^k)\), mean \(k\theta\), and variance \(k\theta^2\). Setting \(k = 1\) recovers the Exponential, and Gamma(\(k, \theta\)) models the sum of \(k\) independent Exponential(\(1/\theta\)) waiting times.
A second cluster of continuous distributions lives naturally on bounded intervals or the positive reals. The Beta distribution on \((0, 1)\) has PDF \(f(p) = p^{\alpha - 1}(1 - p)^{\beta - 1} / B(\alpha, \beta)\), with mean \(\alpha/(\alpha + \beta)\); it is widely used in ML because it serves as the conjugate prior for Bernoulli, Binomial, and Geometric likelihoods, and provides a flexible family for modeling probabilities. The Dirichlet distribution extends Beta to the \(K\)-simplex, with density \(f(\pi) = (1/B(\alpha)) \prod_i \pi_i^{\alpha_i - 1}\) for \(\pi_i \ge 0\) with \(\sum \pi_i = 1\); it is the conjugate prior of the Multinomial and is central to Bayesian topic models. The Log-Normal distribution describes positive random variables whose logarithm is Normal, with mean \(e^{\mu + \sigma^2/2}\), and is appropriate when multiplicative rather than additive variation dominates.
Other continuous families are important in ML for modeling heavy-tailed behavior, sparse signals, or specialized physical processes. The Student's t-distribution has heavier tails than the Normal, with mean 0 (for \(\nu > 1\)) and variance \(\nu/(\nu - 2)\) for \(\nu > 2\), converging to Normal as \(\nu \to \infty\). The Cauchy (or Lorentz) distribution is the t with \(\nu = 1\); it has undefined mean and variance and is a classic heavy-tailed example. The Laplace (double-exponential) distribution \(f(x \mid \mu, b) = (1/(2b)) e^{-|x - \mu|/b}\) has heavier tails than Normal and is often used as a sparsity-inducing prior on regression coefficients. Heavy-tailed distributions in general decay more slowly than exponential — for instance as power laws \(P(X > x) \propto x^{-\alpha}\), exemplified by the Pareto distribution \(f(x) = \alpha x_m^{\alpha} / x^{\alpha + 1}\) for \(x \ge x_m\), which underlies the "80/20 rule" in wealth and city-size modeling, and the discrete Zipf distribution where the \(k\)-th most frequent item has probability proportional to \(1/k^s\). The chi-squared distribution arises as the sum of squared standard normals and is itself a Gamma(\(k/2, 2\)); in ratio form it gives the F-distribution used in ANOVA. Survival and hazard functions provide an alternative language for non-negative distributions: the survival function \(S(t) = P(T > t) = 1 - F(t)\), and the hazard \(h(t) = f(t)/S(t)\) is the instantaneous failure rate conditional on survival. For the Weibull distribution with shape \(k\), the hazard decreases for \(k < 1\) (infant mortality), is constant for \(k = 1\) (Exponential), and increases for \(k > 1\) (wear-out); the Rayleigh distribution, a Weibull with shape 2, models magnitudes of 2D Gaussian vectors, while the Maxwell–Boltzmann distribution, a chi distribution with 3 degrees of freedom, describes molecular speeds.