What is the cumulative distribution function?

The cumulative distribution function (cdf) completely describes the probability distribution of a real-valued random variable, X (Wikipedia 2006). For every real number x, the cdf is given by

Figure 1

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x (Wikipedia 2006). The probability that X lies in the interval (a, b] is therefore F(b) − F(a) if a < b (Wikipedia 2006). 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(Wikipedia 2006).

Note that in the definition above, the "less or equal" sign, '≤' could be replaced with "strictly less" '<' (Wikipedia 2006). This would yield a different function, but either of the two functions can be readily derived from the other (Wikipedia 2006). The only thing to remember is to stick to either definition as mixing them will lead to incorrect results (Wikipedia 2006). In English-speaking countries the convention that uses the weak inequality (≤) rather than the strict inequality (<) is nearly always used (Wikipedia 2006).

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

Figure 2

Example: This example is extracted from (Wikipedia 2006). 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.

Example: This example is extracted from (Wikipedia 2006). 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 (not necessarily strictly) monotone increasing and continuous from the right (right-continuous) (Wikipedia 2006). Furthermore, we have

Figure 3

Figure 4

Every function with these four properties is a cdf (Wikipedia 2006).

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:

Figure 5

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

Figure 6

for all real numbers a and b (Wikipedia 2006).

(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.) (Wikipedia 2006).

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 (Wikipedia 2006).

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 (Wikipedia 2006). The closely related Kuiper's test (pronounced 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 (Wikipedia 2006). 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 (Wikipedia 2006).

References:

Wikipedia. "Cumulative distribution function", Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Cumulative_distribution_function, Last accessed 14 February 2006.

Brandon Hodgson