dichtingsfunctionaaltheorie

The density function, or probability density function (PDF), is a non-negative function that describes the distribution of a continuous random variable. If a variable X has density f, then for any interval [a, b], the probability that X lies in that interval is P(a ≤ X ≤ b) = ∫_a^b f(x) dx. The total probability over the entire real line is 1, so ∫_{−∞}^{∞} f(x) dx = 1. A density function is not itself a probability; a single point has probability zero for continuous distributions.

The cumulative distribution function F(x) = P(X ≤ x) is related by F(x) = ∫_{−∞}^x f(t) dt, and when F is differentiable, f = F'. Densities are defined on a support, the set where f(x) > 0, which can be all real numbers or a finite interval.

Common examples: the normal density f(x) = (1/√(2πσ^2)) exp{−(x−μ)^2/(2σ^2)} for a normal distribution; the uniform density f(x) = 1/(b−a) for x ∈ [a, b] and 0 otherwise; the exponential density f(x) = λ e^{−λx} for x ≥ 0.

Joint densities extend to multiple variables; independence implies f_{X,Y}(x,y) = f_X(x) f_Y(y). Densities are used in estimation (parametric and nonparametric, e.g., kernel density estimation) and in transforming variables via the change-of-variables formula.

A density function differs from a probability mass function, which applies to discrete variables and assigns probabilities to individual outcomes rather than intervals.