Tetthetsfunksjonen

Tetthetsfunksjonen, also known as the density function, is a nonnegative function f that describes the distribution of a continuous random variable X. It satisfies P(a ≤ X ≤ b) = ∫_a^b f(x) dx for any real numbers a < b. The density is with respect to Lebesgue measure, and the corresponding distribution is absolutely continuous. The total probability is 1, so ∫_{-∞}^{∞} f(x) dx = 1. The function itself is not a probability, but a density used to compute probabilities by integration over sets.

If F(x) = P(X ≤ x) is the cumulative distribution function, then F(x) = ∫_{-∞}^x f(t) dt. Conversely, when F is differentiable, its derivative satisfies f(x) = F'(x). Thus the density determines the distribution, and the distribution function determines the density under regularity conditions.

Common examples include: the normal density f(x) = (1/(σ√(2π))) exp(-(x-μ)²/(2σ²)); the exponential density f(x) = λ e^{-λx} for x ≥ 0; and the uniform density f(x) = 1/(b−a) on [a,b]. Many other families exist, such as the Cauchy, Gamma, Beta, and lognormal densities.

Properties and uses: the density is nonnegative, integrates to 1, and probabilities are obtained by integrating f over the desired region. The expected value and higher moments are given by E[X^k] = ∫ x^k f(x) dx when these integrals exist. In higher dimensions, a joint density f(x) on R^n satisfies ∫_{R^n} f(x) dx = 1, with marginals obtained by integration and independence indicated by factorization f(x) = f_X(x) f_Y(y) in the appropriate case.