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Encyclopedia > 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 Probability theory is the mathematical study of probability. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... A random variable is a term used in mathematics and statistics. ...

F(x) = operatorname{P}(Xleq 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 (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. The word probability derives from the Latin probare (to prove, or to test). ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...


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. 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.


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

operatorname{P}(X=b) = F(b) - lim_{x to b^{-}} F(x)

Contents


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(x) = operatorname{P}(X > x) = 1 - F(x).

Examples

As an example, suppose X is uniformly distributed on the unit interval [0, 1]. Then the cdf is given by In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

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.

Properties

Every cumulative distribution function F is (not necessarily strictly) monotone increasing and continuous from the right (right-continuous). Furthermore, we have lim_{xto -infty}F(x)=0 and lim_{xto +infty}F(x)=1. Every function with these four properties is a cdf. Almost all cdfs are cadlag functions. In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a cadlag function is a function f defined on the real numbers (or a subset - for example ) such that its left limit f(t-) and its right limit f(t+) always exist and f(t) = f(t+) for all t. ...


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: In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ...

F(x) = operatorname{P}(Xleq x) = sum_{x_i leq x} operatorname{P}(X = x_i) = sum_{x_i leq x} p(x_i)

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 In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ... Absolute continuity of real functions In mathematics, a real-valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n satisfies... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...

F(b)-F(a) = operatorname{P}(aleq Xleq b) = int_a^b f(x),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. In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...


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 /kœypəʁ/; a bit like "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. In statistics, the Kolmogorov-Smirnov test is used to determine whether two underlying probability distributions differ from each other or whether an underlying probability distribution differs from a hypothesized distribution, in either case based in finite samples. ... In statistics, Kuipers test is closely related to the more well-known Kolmogorov-Smirnov test (or K-S test as it is often called). ...


See also


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