Cumulative distribution function

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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

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

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(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. 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

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

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

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 $\lim_{x\to -\infty}F(x)=0$ and $\lim_{x\to +\infty}F(x)=1$. Every function with these four properties is a cdf. Almost all cdfs are cadlag functions.

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

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

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

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

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.

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.

See also

• Descriptive statistics
• Probability distribution
• Probability density function
• Empirical distribution function

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