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In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Y = g( X ) is a continuous differentiable function, then the density of Y can be written as
The density varies with temperature, but not linearly: as the temperature increases, the density rises to a peak at 3.98 °C (39.16 °F) and then decreases; [33] the initial increase is unusual because most liquids undergo thermal expansion so that the density only decreases as a function of temperature.
Its density is infinitely differentiable, indeed supersmooth of order 2. [23] Furthermore, the density of the standard normal distribution (i.e. = and =) also has the following properties: Its first derivative is ′ = ().
If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There ...
The ratio of two extensive properties of the same object or system is an intensive property. For example, the ratio of an object's mass and volume, which are two extensive properties, is density, which is an intensive property. [10] More generally properties can be combined to give new properties, which may be called derived or composite ...
The connection to the density pair distribution function was given by the Ornstein–Zernike equation. The importance of correlation for thermodynamic properties was explored through density distribution functions. The functional derivative was introduced to define the distribution functions of classical mechanical systems. Theories were ...
The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. This value is widely used to investigate various physical properties of matter.