Key Continuous Distributions

The mathematics of uncertainty

Beyond the Gaussian, a few continuous distributions show up again and again. Each answers a different kind of question: "anywhere in a range?", "how long until the next event?", "how is an unknown probability itself distributed?"

Uniform U(a, b) spreads probability flatly across an interval, with constant density 1/(b−a). It's the default "I know nothing except the range," and the raw material for sampling: every random generator starts from U(0,1).

Exponential(λ) models the time until a random event when events happen at a steady average rate λ. It's memoryless: having waited a while doesn't change the time still to wait.

Where this lives in MLDirichlet is a "distribution over distributions": it generates the mixing weights in topic models (LDA) and the category probabilities a Bayesian classifier averages over. Beta is the go-to prior for a probability you're estimating, like a click-through rate or a coin's bias, and it powers Thompson sampling in bandits. The Uniform is the entropy source every other sampler transforms.

▶ Key Continuous Distributions

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