Fungsi sebaran kumulatif
nu mana sisi beulah katuhu ngagambarkeun probabilitas nu mana variabel acak X dicokot tina nilai nu kurang tina atawa sarua jeung x. Probabilitas X aya dina interval (a, b] nyaéta F(b) − F(a) lamun a < b. Geus ilahar ngagunakeun huruf F gedé keur fungsi sebaran kumulatif, nu jelas béda jeung huruf f leutik nu dipaké keur fungsi dénsitas probabilitas jeung probability mass function.
Fungsi sebaran kumulatif X bisa dihartikeun dina watesan fungsi dénsitas probabilitas f saperti kieu:
Note that in the definition above, the "less or equal" sign, '≤' is a convention, but it is an important and universally used one. The proper use of tables of the Binomial and Poisson distributions depend upon this convention. Moréover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation.
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:
for all réal 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 "≤" céases 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 édit
The "point probability" that X is exactly b can be found as
Kolmogorov-Smirnov and Kuiper's tests édit
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 idéal distribution. The closely related Kuiper's test (pronounced /kœypəʁ/) 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 yéar or if sales of a product vary by day of the week or day of the month.
Complementary cumulative distribution function édit
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
In survival analysis, is called the survival function and denoted .
As an example, suppose X is uniformly distributed on the unit interval [0, 1]. Then the CDF of X is given by
Take another example, suppose X takes only the discrete values 0 and 1, with equal probability. Then the CDF of X is given by
If the cdf F is strictly incréasing and continuous then is the unique réal number such that .
Unfortunately, the distribution does not, in general, have an inverse. One may define, for ,
Example 1: The median is .
Example 2: Put . Then we call the 95th percentile.
The inverse of the cdf is called the quantile function.