probabilitydensity

Probability density, often called a probability density function (pdf), is a function used to specify the distribution of a continuous random variable. If X is such a variable with 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 density itself is not a probability; it can take values greater than 1, as long as the total area under the curve over the entire real line equals 1.

A valid density satisfies two key conditions: f(x) ≥ 0 for all x, and the integral of f over the entire real line equals 1, i.e., ∫_{-∞}^{∞} f(x) dx = 1. Densities can be defined in higher dimensions, where f(x) ≥ 0 on R^n and ∫_{R^n} f(x) dx = 1. The probability of a region is the integral of the density over that region. The cumulative distribution function F(x) is related by F(x) = P(X ≤ x) = ∫_{-∞}^x f(t) dt.

Common examples include the standard normal density f(x) = (1/√(2π)) e^{-x^2/2}, which integrates to 1, and the uniform density f(x) = 1/(b−a) on [a, b] and 0 outside. Densities enable computation of expectations: E[g(X)] = ∫ g(x) f(x) dx. They also transform under changes of variables via the Jacobian, and marginals are obtained by integrating joint densities over the unwanted variables.

---