Continuous Probability Distributions
Continuous distributions model quantities that can take any value in an interval: heights, waiting times, measurement errors. Many arise as limits of discrete models or from natural symmetry assumptions.
Continuous Uniform Distribution
The uniform distribution on [a, b] has density f(x) = 1/(b − a) for x in the interval. Mean is (a + b)/2 and variance is (b − a)2/12. It is the natural model for no-preference situations and the basis of simulation: most random number generators produce uniform pseudorandom numbers in [0, 1].
Exponential Distribution
The exponential distribution with rate λ has density f(x) = λ e−λx for x ≥ 0. Mean is 1/λ and variance is 1/λ2. Like the geometric, it is memoryless. It models waiting times between events of a Poisson process, time to failure of memoryless components, and interarrival times in queueing theory.
Gamma Distribution
The gamma distribution Gamma(k, θ) generalizes the exponential. It is the sum of k independent exponentials with mean θ. Applications include time to the kth event in a Poisson process and prior distributions in Bayesian analysis. Beta and chi-square distributions are special cases.
Normal Distribution
The normal (Gaussian) distribution N(μ, σ2) has the familiar bell-shaped density f(x) = 1/(σ √(2π)) e−(x − μ)2/(2σ2). It is symmetric about its mean μ, and the empirical rule states that about 68%, 95%, and 99.7% of the area lies within 1, 2, and 3 standard deviations of μ. Standardizing Z = (X − μ)/σ converts any normal to N(0, 1).
Central Limit Theorem
The central limit theorem (CLT) states that the sum (or mean) of a large number of independent random variables with finite variance is approximately normal, regardless of the original distribution. This is why the normal appears so ubiquitously; it justifies the normal approximation and underlies classical hypothesis testing.
Lognormal Distribution
If log X is normal, then X is lognormal. It models quantities that are strictly positive and multiplicatively determined: stock prices, file sizes, income distributions. Lognormal is right-skewed.
Chi-Square, t, and F Distributions
The chi-square distribution χ2(k) is the sum of k squared standard normals; it is used in variance tests. Student's t distribution arises from normal observations with estimated variance; it governs confidence intervals for small samples. The F distribution is a ratio of chi-squares and is central in ANOVA.
Beta Distribution
The beta distribution Beta(α, β) on [0, 1] is widely used as a Bayesian prior for probabilities. Its shape adapts from uniform (α = β = 1) to skewed and bimodal forms.
Simulation and Inverse Transform
Continuous random variates are generated from uniforms by inverse transform sampling: if U is uniform on [0, 1], then F−1(U) has CDF F. Rejection sampling and the Box–Muller method are alternatives when F−1 is not easy to compute.
Summary
Continuous distributions model measurements and waiting times. The normal distribution dominates theory and practice through the central limit theorem; exponential, gamma, and chi-square capture special structures. Understanding each distribution's shape, moments, and typical use is essential for statistical modelling.