Fungsi sebaran kumulatif

Luncat ka: pituduh, sungsi

Dina matematik, fungsi distribusi kumulatip (disingkat cdf) ngajelaskeun probability distribution ti sakabéh nilai-real variabel acak, X. Keur satiap réal number x, cdf dirumuskeun ku

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

numana sisi beulah katuhu ngagambarkeun probabilitas numana variabel acak X dicokot tina nilai nu kurang tina atawa sarua jeung x. Probabilitas X aya dina interval (ab] nyaéta F(b) − F(a) lamun a < b. Geus ilahar ngagunakeun huruf F gede keur fungsi sebaran kumulatif, nu jelas beda 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:

${\displaystyle F(x)=\int _{-\infty }^{x}f(t)\,dt}$

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.

Properties

From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part.

Every cumulative distribution function F is (not necessarily strictly) monotone increasing and right-continuous. Furthermore, we have

${\displaystyle \lim _{x\to -\infty }F(x)=0,\quad \lim _{x\to +\infty }F(x)=1.}$

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:

${\displaystyle F(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname {P} (X=x_{i})=\sum _{x_{i}\leq x}p(x_{i})}$

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

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

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

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

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

Kolmogorov-Smirnov and Kuiper's tests

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

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

${\displaystyle F_{c}(x)=\operatorname {P} (X>x)=1-F(x)}$.

In survival analysis, ${\displaystyle F_{c}(x)}$ is called the survival function and denoted ${\displaystyle S(x)}$.

Contohna

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

${\displaystyle F(x)={\begin{cases}0&:\ x<0\\x&:\ 0\leq x\leq 1\\1&:\ 1

Take another example, suppose X takes only the discrete values 0 and 1, with equal probability. Then the CDF of X is given by

${\displaystyle F(x)={\begin{cases}0&:\ x<0\\1/2&:\ 0\leq x<1\\1&:\ 1\leq x\end{cases}}}$

Inverse

If the cdf F is strictly incréasing and continuous then ${\displaystyle F^{-1}(y),y\in [0,1]}$ is the unique réal number ${\displaystyle x}$ such that ${\displaystyle F(x)=y}$.

Unfortunately, the distribution does not, in general, have an inverse. One may define, for ${\displaystyle y\in [0,1]}$,

${\displaystyle F^{-1}(y)=\inf _{r\in \mathbb {R} }\{F(r)>y\}}$.

Example 1: The median is ${\displaystyle F^{-1}(0.5)}$.

Example 2: Put ${\displaystyle \tau =F^{-1}(0.95)}$. Then we call ${\displaystyle \tau }$ the 95th percentile.

The inverse of the cdf is called the quantile function.