Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable

X

{\displaystyle X}

, or just distribution function of

X

{\displaystyle X}

, evaluated at

x

{\displaystyle x}

, is the probability that

X

{\displaystyle X}

will take a value less than or equal to

x

{\displaystyle x}

.

Every probability distribution defined on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function)

F

:

R

→

[

0

,

1

]

{\displaystyle F\colon \mathbb {R} \rightarrow [0,1]}

satisfying

lim

x

→

−

∞

F

(

x

)

=

0

{\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}

and

lim

x

→

∞

F

(

x

)

=

1

{\displaystyle \lim _{x\rightarrow \infty }F(x)=1}

. And conversely, any such function

F

{\displaystyle F}

is the cumulative distribution function of such a probability distribution.

In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to

x

{\displaystyle x}

. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

Definition

The cumulative distribution function of a real-valued random variable

X

{\displaystyle X}

is the function given by

where the right-hand side represents the probability that the random variable

X

{\displaystyle X}

takes on a value less than or equal to

x

{\displaystyle x}

.

The probability that

X

{\displaystyle X}

lies in the semi-closed interval

(

a

,

b

]

{\displaystyle (a,b]}

, where

a

<

b

{\displaystyle a<b}

, is therefore

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 the distinction is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. Moreover, important formulas like Paul Lévy's inversion formula for the characteristic function also rely on the "less than or equal" formulation.