Cumulative distribution function

In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by

<math>F(x) = \operatorname{P}(X\leq x),<math>

where the right-hand side represents the probability that the variable X takes on a value less than or equal to x. The probability that X lies in the interval (ab) is therefore F(b) − F(a) if a ≤ b. 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.

Note that in the definition above, the "less or equal" sign, '≤' could be replaced with "strictly less" '<'. This would yield a different function, but either of the two functions can be readily derived from the other. One could even use "greater" sign there (changing cdf properties even more). The only thing to remember is to stick to either definition as mixing them will lead to incorrect results. In English-speaking countries the convention that uses the weak inequality (≤) rather than the strict inequality (<) is nearly always used.


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

F(x) = 0, if x < 0;
F(x) = x, if 0 ≤ x ≤ 1;
F(x) = 1, if x > 1.

For a different example, suppose X takes only the values 0 and 1, with equal probability. Then the cdf is given by

F(x) = 0, if x < 0;
F(x) = 1/2, if 0 ≤ x < 1;
F(x) = 1, if x ≥ 1.


Every cumulative distribution function F is monotone increasing and continuous from the right. Furthermore, we have <math>\lim_{x\to -\infty}F(x)=0<math> and <math>\lim_{x\to +\infty}F(x)=1<math>. Every function with these four properties is a cdf.

If X is a discrete random variable, then it attains values x1, x2, ... with probability p1, p2 etc., and the cdf of X will be discontinuous at the points xi and constant in between.

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

<math>F(b)-F(a) = \operatorname{P}(a\leq X\leq b) = \int_a^b f(x)\,dx<math>

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.

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 (pronounced in Dutch the way an Cowper might be pronounced in English) 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.

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