Tiheysfunktion

Tiheysfunktion, or density function, is a mathematical description of how probability mass is distributed over the real numbers for a continuous random variable. For a random variable X, a density function f satisfies f(x) ≥ 0 for all x and the integral over the entire real line equals 1. The probability that X falls within an interval (a, b) is given by the integral ∫a^b f(x) dx. The cumulative distribution function F(x) = P(X ≤ x) is the integral of f from -∞ to x, and where F is differentiable, f = F'.

Density functions are defined with respect to Lebesgue measure; not every distribution has a density. Discrete distributions are described by a probability mass function rather than a density, and mixed distributions may have a density with respect to Lebesgue measure plus atoms. Densities are unique almost everywhere: if f and g are densities for the same distribution, f = g almost everywhere.

Examples: the standard normal density f(x) = (1/√(2π)) e^{-x^2/2}; the uniform density on [0, 1]: f(x) = 1 for 0 ≤ x ≤ 1 and 0 otherwise; the exponential density f(x) = λ e^{-λ x} for x ≥ 0. Densities enable calculations of expectations, variances, and probabilities, and they underpin many statistical methods.

Transformation: if Y = g(X) is a monotone function, the density of Y is f_Y(y) = f_X(g^{-1}(y)) · |d/dy g^{-1}(y)|. Densities are fundamental in inference, physics, and engineering, and they are often estimated from data using parametric or nonparametric methods.

See also: probability density function, cumulative distribution function, Lebesgue measure.