Definitions

# Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF), also probability distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X. For every real number x, the CDF of X is given by

$x mapsto F_X\left(x\right) = operatorname\left\{P\right\}\left(Xleq x\right),$

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 (ab] is therefore $F_X\left(b\right)-F_X\left(a\right)$ if a < b.

If treating several random variables XY, ... 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:

$F\left(x\right) = int_\left\{-infty\right\}^x f\left(t\right),dt$

Note that in the definition above, the "less or equal" sign, '≤' is a convention, but it is a universally used one, and 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.

## Properties

Every cumulative distribution function F is (not necessarily strictly) monotone increasing and right-continuous. Furthermore, we have

$lim_\left\{xto -infty\right\}F\left(x\right)=0, quad lim_\left\{xto +infty\right\}F\left(x\right)=1.$

Every function with these four properties is a cdf. The properties imply that all CDFs are càdlàg functions.

If X is a discrete random variable, then it attains values x1, x2, ... with probability pi = P(xi), and the cdf of X will be discontinuous at the points xi and constant in between:

$F\left(x\right) = operatorname\left\{P\right\}\left(Xleq x\right) = sum_\left\{x_i leq x\right\} operatorname\left\{P\right\}\left(X = x_i\right) = sum_\left\{x_i leq x\right\} p\left(x_i\right)$

If the CDF F of X is continuous, then X is a continuous random variable; if furthermore F is absolutely continuous, then there exists a Lebesgue-integrable function f(x) such that

$F\left(b\right)-F\left(a\right) = operatorname\left\{P\right\}\left(aleq Xleq b\right) = int_a^b f\left(x\right),dx$

for all real numbers a and b. (The first of the two equalities displayed above would not be correct in general if we had not said that the distribution is continuous. Continuity of the distribution implies that P (X = a) = P (X = b) = 0, so the difference between "<" and "≤" ceases to be important in this context.) The function f is equal to the derivative of F almost everywhere, and it is called the probability density function of the distribution of X.

### Point probability

The "point probability" that X is exactly b can be found as

$operatorname\left\{P\right\}\left(X=b\right) = F\left(b\right) - lim_\left\{x to b^\left\{-\right\}\right\} F\left(x\right)$

## Kolmogorov-Smirnov and Kuiper's tests

The Kolmogorov-Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper's test is useful if the domain of the distribution is cyclic as in day of the week. For instance we might use Kuiper's test to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.

## Complementary cumulative distribution function

Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function (ccdf), defined as

$F_c\left(x\right) = operatorname\left\{P\right\}\left(X > x\right) = 1 - F\left(x\right)$.

In survival analysis, $F_c\left(x\right)$ is called the survival function and denoted $S\left(x\right)$.

## Folded cumulative distribution

While the plot of a cumulative distribution often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median and dispersion of the distribution or of the empirical results.

## Examples

As an example, suppose X is uniformly distributed on the unit interval [0, 1]. Then the CDF of X is given by

$F\left(x\right) = begin\left\{cases\right\}$
0 &: x < 0 x &: 0 le x le 1 1 &: 1 < x end{cases}

Take another example, suppose X takes only the discrete values 0 and 1, with equal probability. Then the CDF of X is given by

$F\left(x\right) = begin\left\{cases\right\}$
0 &: x < 0 1/2 &: 0 le x < 1 1 &: 1 le x end{cases}

## Inverse

If the cdf F is strictly increasing and continuous then $F^\left\{-1\right\}\left(y \right), y in \left[0,1\right]$ is the unique real number $x$ such that $F\left(x\right) = y$.

Unfortunately, the distribution does not, in general, have an inverse. One may define, for $y in \left[0,1\right]$,


F^{-1}(y) = inf_{x in mathbb{R}} { F(x) geq y } .

Example 1: The median is $F^\left\{-1\right\}\left(0.5 \right)$.

Example 2: Put $tau = F^\left\{-1\right\}\left(0.95 \right)$. Then we call $tau$ the 95th percentile.

The inverse of the cdf is called the quantile function.

The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions. Some useful properties of the inverse cdf are:

1. $F^\left\{-1\right\}$ is nondecreasing
2. $F^\left\{-1\right\}\left(F\left(x\right)\right) leq x$
3. $F\left(F^\left\{-1\right\}\left(y\right)\right) geq y$
4. $F^\left\{-1\right\}\left(y\right) leq x$ if and only if $y leq F\left(x\right)$
5. If $Y$ has a $U\left[0, 1\right]$ distribution then $F^\left\{-1\right\}\left(Y\right)$ is distributed as $F$
6. If $\left\{X_alpha\right\}$ is a collection of independent $F$-distributed random variables defined on the same sample space, then there exist random variables $Y_alpha$ such that $Y_alpha$ is distributed as $U\left[0,1\right]$ and $F^\left\{-1\right\}\left(Y_alpha\right) = X_alpha$ with probability 1 for all $alpha$.

## Multivariate Case

When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables X,Y, the joint CDF is given by

$x,y to F\left(x,y\right) = operatorname\left\{P\right\}\left(Xleq x,Yleq y\right),$

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x and that Y takes on a value less than or equal to y