دالة التوزيع التراكمي

Cumulative distribution function for the exponential distribution

Cumulative distribution function for the normal distribution

في نظرية الاحتمالات والإحصاء، دالة التوزيع التراكمي (الإنكليزية: cumulative distribution function واختصاراً CDF)، هو تابع يصف احتمال أن يأخذ المتغير العشوائي X ضمن توزيع احتمالي معطى قيمة أصغر أو تساوي x.

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 تعريف

For every real number x, the CDF of a real-valued random variable X is given by

$x\mapsto F_{X}(x)=\operatorname {P} (X\leq x),$

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x. The probability that X lies in the interval (a, b] is therefore $F_{X}(b)-F_{X}(a)$ if a < b.

If treating several random variables X, Y, ... etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is omitted. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution.

The CDF of X can be defined in terms of the probability density function ƒ as follows:

$F(x)=\int _{-\infty }^{x}f(t)\,dt.$

Note that in the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depend upon this convention. Moreover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation.

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