integratiedensity

Integrated density refers to the cumulative quantity obtained by integrating a density function. In mathematics, if f is a density on the real line, its integrated density F is defined by F(x) = ∫_{-∞}^{x} f(t) dt. When f is a probability density, F is the associated cumulative distribution function (CDF) of the random variable.

Key properties: F is nondecreasing and right-continuous, with F(-∞) = 0 and F(∞) = 1 if f integrates to 1. On a finite interval [a, b], the integrated density up to x ∈ [a, b] is F(x) = ∫_{a}^{x} f(t) dt. If f ≥ 0 and is integrable to 1, F is a distribution function, and where f is continuous, F′(x) = f(x).

Examples: For a uniform distribution on [0, 1], f(x) = 1 for x ∈ [0, 1], and F(x) = 0 for x ≤ 0, F(x) = x for 0 ≤ x ≤ 1, and F(x) = 1 for x ≥ 1. For the normal distribution, F is the standard normal CDF, the integral of its density φ.

Applications: Integrated densities are used to compute probabilities, quantiles, and expectations in statistics. In physics and spectral theory, a closely related concept is the integrated density of states (IDOS), which counts the number of quantum states with energy not exceeding a threshold per unit volume by integrating the density of states.

Notes: The term is general and may be referred to as cumulative density or accumulated density depending on context. Related concepts include the density function, the cumulative distribution function, and density of states.