# Distribution Function

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## distribution function

[‚dis·trə′byü·shən ‚fəŋk·shən]
(industrial engineering)

## Distribution Function

a fundamental concept of statistical mechanics. In classical statistical mechanics the distribution function characterizes the probability density of the distribution of the particles of a statistical system in phase space—that is, with respect to the coordinates qi and momenta pi. In quantum statistical mechanics the distribution function characterizes the probability of a distribution over quantum-mechanical states.

In classical statistical mechanics the distribution function f(p, q, t) defines the probability dw = f(p, q, t) dpdq of finding a system of N particles at a time t in the volume element of phase space dpdq = dp1dq1 . . . dpNdqN around the point p1, q1, . . ., pN, qN. Since the transposition of identical particles does not change the state, the phase volume should be reduced by a factor of N!. Furthermore, it is convenient to convert to a dimensionless volume element of phase space by replacing dpdq with dpdq/N!h3N, where Planck’s constant h determines the minimum cell size in phase space. (See alsoGIBBS DISTRIBUTION.)

References in periodicals archive ?
There is the distribution function (14) of the exponential distribution.
where [(X).sub.[alpha][beta]] = <a[absolute value of (X)][beta]> for electron states a and p, is the I component of the electron position vector, jk is the k component of the single electron current density operator, fa is the Fermi distribution function for an electron with energy [E.sub.[alpha]], and [E.sub.[alpha][beta]] = [E.sub.[alpha]] - [E.sub.[beta]].
The conditional cumulative distribution function (CCDF) of outage ([F.sup.c.sub.out]) can be expressed as
On the basis of the theory of probability and the relationship of marginal probability distribution function, conditional probability distribution function, and integrated probability distribution function, we construct an integrated cost and schedule risk estimation model.
The study on the measurement of income concentration ratio conducted by Lorenz  (1905)was regarded as a revolution in economics and statistics, resulting in tens of thousands of papers published in magazines about statistics and econometrics and starting the research on the relationship between the Distribution Function and income inequality.
It is common in practice to consider only the lowest-order moments of the distribution function, which are directly related to density, charge flux, kinetic energy, heat flux, and so on.
The Weibull distribution function [23, 24] is the most widely used statistical function for the characterization of strength distribution of advanced ceramics.
The estimation of unknown particle shape distribution function Fs based on the estimator of Ft is an ill-posed problem.
and to derive an evolution equation for the distribution function. This can be done by using the macroscopic balance of mass and the mesoscopic balance of mass.
Here Y is the result of mapping the random variable x through its own Cumulative Distribution Function. The cumulative distribution of Y can be easily computed.
In order to define a thickness distribution function for further study, topology optimization for the shell geometry obtained with constant thickness optimization was carried out.
where F(t) is the cumulative distribution function and [tau](t) = f(t)/F(t) is the reversed hazard function or reversed failure rate of X.

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