Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,^[1] completely describes the probability distribution of a real-valued random variable X. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way For every real number x, the CDF of X is given by

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

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x. Probability is the likelihood or chance that something is the case or will happen The probability that X lies in the interval (a, b] is therefore $F X (b) - F X (a)$ if a < b. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

If treating several random variables X,Y,. . . etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is omitted. 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. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In Probability theory, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete Random variable This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the Normal Distribution. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields

The CDF of X can be defined in terms of the probability density function f as follows:

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

Note that in the definition above, the "less or equal" sign, '≤' is a convention, but it is a universally used one, and is important for discrete distributions. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability The proper use of tables of the Binomial and Poisson distributions depend upon this convention. Moreover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation.

1 Properties
- 1.1 Point probability
2 Kolmogorov-Smirnov and Kuiper's tests
3 Complementary cumulative distribution function
4 Folded cumulative distribution
5 Examples
6 Inverse
7 Multivariate Case
8 See also
9 References
10 External links

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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Furthermore, we have

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

Every function with these four properties is a cdf. The properties imply that all CDFs are càdlàg functions. In Mathematics, a càdlàg (French "continue à droite limitée à gauche" RCLL ("right continuous with left limits" or corlol

If X is a discrete random variable, then it attains values x₁, x₂, . In Probability theory, a Probability distribution is called discrete if it is characterized by a Probability mass function. . . 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Probability theory, a Probability distribution is called continuous if its Cumulative distribution function is continuous. In Mathematics, one may talk about absolute continuity of functions and absolute continuity of measures, and these two notions are closely connected In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of (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 Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change 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 In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability

Point probability

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

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

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 ideal distribution. In Statistics, the Kolmogorov &ndash Smirnov test (also called the K-S test for brevity is a form of Minimum distance estimation used The closely related Kuiper's test (pronounced [kœypəʁ]) is useful if the domain of the distribution is cyclic as in day of the week. In Statistics, Kuiper's test is closely related to the more well-known Kolmogorov-Smirnov test (or K-S test as it is often called 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.

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

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

In survival analysis, $F c (x)$ is called the survival function and denoted $S (x)$ . Survival analysis is a branch of Statistics which deals with death in biological organisms and failure in mechanical systems The survival function, also known as a survivor function or reliability function, is a property of any Random variable that maps a set of events

Folded cumulative distribution

Example of the folded cumulative distribution for a normal distribution function

While a the plot of a cumulative distibution often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over,^[2] thus using two scales, one for the upslope and another for the downslope. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields This form of illustration emphasises the median and dispersion of the distribution or of the empirical results. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Statistics, (statistical dispersion (also called statistical variability or variation) is variability or spread in a Variable or a Probability

Examples

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

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

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

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

Inverse

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

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

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

Example 1: The median is $F - 1 (0. 5)$ .

Example 2: Put $τ = F - 1 (0. 95)$ . Then we call $τ$ the 95th percentile.

The inverse of the cdf is called the quantile function. See also Quantile. In Probability theory, a quantile function of a Probability distribution is the inverse

Multivariate Case

When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables X,Y, the joint CDF is given by

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

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x and that Y takes on a value less than or equal to y

References

^ Eric W. Weisstein, Distribution Function at MathWorld. Probability is the likelihood or chance that something is the case or will happen Descriptive Statistics are used to describe the basic features of the Data gathered from an experimental study in various ways In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/ n at each of the Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value In Statistics, a Q-Q plot ( "Q" stands for Quantile) is a graphical method for diagnosing differences between the Probability An ogive ("Oh-jive" is a curved shape figure or feature See also Quantile. In Probability theory, a quantile function of a Probability distribution is the inverse Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
^ Medcalc.be: Mountain plot

External links

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents