
In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics, the cumulative distribution function (CDF) of a real-valued
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
, or just distribution function of
, evaluated at
, is the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
that
will take a value less than or equal to
.
Every probability distribution
supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous''
''monotonic increasing'' cumulative distribution function
continuous distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, it gives the area under the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
from minus infinity to
x. Cumulative distribution functions are also used to specify the distribution of
multivariate random variable
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
s.
Definition
The cumulative distribution function of a real-valued
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
X is the function given by
where the right-hand side represents the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
that the random variable
X takes on a value less than or equal to
x.
The probability that
X lies in the semi-closed
interval (a,b], where
a < b, is therefore
[
In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the Binomial distribution, binomial and ]Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known ...
s depends upon this convention. Moreover, important formulas like Paul Lévy's inversion formula for the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
also rely on the "less than or equal" formulation.
If treating several random variables X, Y, \ldots etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually 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 function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
s and probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
s. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
uses \Phi and \phi instead of F and f, respectively.
The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating using the Fundamental Theorem of Calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
; i.e. given F(x),
f(x) = \frac
as long as the derivative exists.
The CDF of a continuous random variable
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
X can be expressed as the integral of its probability density function f_X as follows:[
F_X(x) = \int_^x f_X(t) \, dt.
In the case of a random variable X which has distribution having a discrete component at a value b,
\operatorname(X=b) = F_X(b) - \lim_ F_X(x).
If F_X is continuous at b, this equals zero and there is no discrete component at b.
]
Properties
Every cumulative distribution function F_X is non-decreasing
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
[ and ]right-continuous
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
,[ which makes it a ]càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subse ...
function. Furthermore,
\lim_ F_X(x) = 0, \quad \lim_ F_X(x) = 1.
Every function with these four properties is a CDF, i.e., for every such function, a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
can be defined such that the function is the cumulative distribution function of that random variable.
If X is a purely discrete random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, then it attains values x_1,x_2,\ldots with probability p_i = p(x_i), and the CDF of X will be discontinuous at the points x_i:
F_X(x) = \operatorname(X\leq x) = \sum_ \operatorname(X = x_i) = \sum_ p(x_i).
If the CDF F_X of a real valued random variable X is continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
, then X is a continuous random variable
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
; if furthermore F_X is absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
, then there exists a Lebesgue-integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
function f_X(x) such that
F_X(b)-F_X(a) = \operatorname(a< X\leq b) = \int_a^b f_X(x)\,dx
for all real numbers a and b. The function f_X is equal to the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of F_X almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
, and it is called the probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
of the distribution of X.
If X has finite L1-norm
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki ...
, that is, the expectation of , X, is finite, then \mathbb E = \int_^\infty t dF_X(t)and for any x \geq 0,
\begin
x (1-F_X(x)) & \leq \int_x^ t dF_X(t) \\
x F_X(-x) & \leq \int_^ (-t) dF_X(t)
\endas shown in the diagram.
In particular, we have \lim_ x F(x) = 0, \quad \lim_ x (1-F(x)) = 0.
Examples
As an example, suppose X is uniformly distributed on the unit interval ,1/math>.
Then the CDF of X is given by
F_X(x) = \begin
0 &:\ x < 0\\
x &:\ 0 \le x \le 1\\
1 &:\ x > 1
\end
Suppose instead that X takes only the discrete values 0 and 1, with equal probability.
Then the CDF of X is given by
F_X(x) = \begin
0 &:\ x < 0\\
1/2 &:\ 0 \le x < 1\\
1 &:\ x \ge 1
\end
Suppose X is exponential distributed. Then the CDF of X is given by
F_X(x;\lambda) = \begin
1-e^ & x \ge 0, \\
0 & x < 0.
\end
Here ''λ'' > 0 is the parameter of the distribution, often called the rate parameter.
Suppose X is normal distributed. Then the CDF of X is given by
F(x;\mu,\sigma) = \frac \int_^x \exp \left( -\frac \right)\, dt.
Here the parameter \mu is the mean or expectation of the distribution; and \sigma is its standard deviation.
A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the standard normal table A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, which are the values of the cumulative distribution function of the normal distribution. It is used to find the probability that a ...
, the unit normal table, or the Z table.
Suppose X is binomial distributed. Then the CDF of X is given by
F(k;n,p) = \Pr(X\leq k) = \sum _^ p^ (1-p)^
Here p is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of n independent experiments, and \lfloor k\rfloor is the "floor" under k, i.e. the greatest integer less than or equal to k.
Derived functions
Complementary cumulative distribution function (tail distribution)
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) or simply the tail distribution or exceedance, and is defined as
\bar F_X(x) = \operatorname(X > x) = 1 - F_X(x).
This has applications in statistical
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
ing, for example, because the one-sided p-value
In null-hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small ''p''-value means ...
is the probability of observing a test statistic ''at least'' as extreme as the one observed. Thus, provided that the test statistic
A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specifie ...
, ''T'', has a continuous distribution, the one-sided p-value
In null-hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small ''p''-value means ...
is simply given by the ccdf: for an observed value t of the test statistic
p= \operatorname(T \ge t) = \operatorname(T > t) = 1 - F_T(t).
In survival analysis
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysi ...
, \bar F_X(x) is called the survival function
The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time.
The survival function is also known as the survivor function
or reliability function.
The te ...
and denoted S(x) , while the term ''reliability function'' is common in engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
.
;Properties
* For a non-negative continuous random variable having an expectation, Markov's inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Marko ...
states that \bar F_X(x) \leq \frac .
* As x \to \infty, \bar F_X(x) \to 0 , and in fact \bar F_X(x) = o(1/x) provided that \operatorname(X) is finite.
Proof:
Assuming X has a density function f_X, for any c > 0
\operatorname(X) = \int_0^\infty x f_X(x) \, dx \geq \int_0^c x f_X(x) \, dx + c\int_c^\infty f_X(x) \, dx
Then, on recognizing \bar F_X(c) = \int_c^\infty f_X(x) \, dx and rearranging terms,
0 \leq c\bar F_X(c) \leq \operatorname(X) - \int_0^c x f_X(x) \, dx \to 0 \text c \to \infty
as claimed.
* For a random variable having an expectation, \operatorname(X) = \int_0^\infty \bar F_X(x) \, dx - \int_^0 F_X(x) \, dx and for a non-negative random variable the second term is 0.
If the random variable can only take non-negative integer values, this is equivalent to \operatorname(X) = \sum_^\infty \bar F_X(n).
Folded cumulative distribution
While the plot of a cumulative distribution F 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,[
] that is
:F_\text(x)=F(x)1_+(1-F(x))1_
where 1_ denotes the indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
and the second summand is the survivor function, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median, dispersion
Dispersion may refer to:
Economics and finance
* Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
* Wage dispersion, the amount of variat ...
(specifically, the mean absolute deviation
The average absolute deviation (AAD) of a data set is the average of the Absolute value, absolute Deviation (statistics), deviations from a central tendency, central point. It is a summary statistics, summary statistic of statistical dispersion or ...
from the median) and skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimo ...
of the distribution or of the empirical results.
Inverse distribution function (quantile function)
If the CDF ''F'' is strictly increasing and continuous then F^( p ), p \in ,1 is the unique real number x such that F(x) = p . This defines the inverse distribution function or quantile function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value e ...
.
Some distributions do not have a unique inverse (for example if f_X(x)=0 for all a, causing F_X to be constant). In this case, one may use the generalized inverse distribution function, which is defined as
:
F^(p) = \inf \, \quad \forall p \in ,1
* Example 1: The median is F^( 0.5 ).
* Example 2: Put \tau = F^( 0.95 ) . Then we call \tau the 95th percentile.
Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:
# F^ is nondecreasing
# F^(F(x)) \leq x
# F(F^(p)) \geq p
# F^(p) \leq x if and only if p \leq F(x)
# If Y has a U, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> distribution then F^(Y) is distributed as F. This is used in random number generation
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular outc ...
using the inverse transform sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden ruleAalto University, N. Hyvönen, Computational methods in inverse probl ...
-method.
# If \ is a collection of independent F-distributed random variables defined on the same sample space, then there exist random variables Y_\alpha such that Y_\alpha is distributed as U ,1/math> and F^(Y_\alpha) = X_\alpha with probability 1 for all \alpha.
The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.
Empirical distribution function
The empirical distribution function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function ...
is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function.
Multivariate case
Definition for two random variables
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 F_ is given by[
where the right-hand side represents the ]probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
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.
Example of joint cumulative distribution function:
For two continuous variables ''X'' and ''Y'': \Pr(a < X < b \text c < Y < d) = \int_a^b \int_c^d f(x,y) \, dy \, dx;
For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of ''X'' and ''Y'', and here is the example:
given the joint probability mass function in tabular form, determine the joint cumulative distribution function.
Solution: using the given table of probabilities for each potential range of ''X'' and ''Y'', the joint cumulative distribution function may be constructed in tabular form:
Definition for more than two random variables
For N random variables X_1,\ldots,X_N, the joint CDF F_ is given by
Interpreting the N random variables as a random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
\mathbf = (X_1, \ldots, X_N)^T yields a shorter notation:
F_(\mathbf) = \operatorname(X_1 \leq x_1,\ldots,X_N \leq x_N)
Properties
Every multivariate CDF is:
# Monotonically non-decreasing for each of its variables,
# Right-continuous in each of its variables,
# 0\leq F_(x_1,\ldots,x_n)\leq 1,
# \lim_F_(x_1,\ldots,x_n)=1 \text \lim_F_(x_1,\ldots,x_n)=0, \text i.
Any function satisfying the above four properties is not a multivariate CDF, unlike in the single dimension case. For example, let F(x,y)=0 for x<0 or x+y<1 or y<0 and let F(x,y)=1 otherwise. It is easy to see that the above conditions are met, and yet F is not a CDF since if it was, then \operatorname\left(\frac < X \leq 1, \frac < Y \leq 1\right)=-1 as explained below.
The probability that a point belongs to a hyperrectangle
In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions.
A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all ...
is analogous to the 1-dimensional case:
F_(a, c) + F_(b, d) - F_(a, d) - F_(b, c) = \operatorname(a < X_1 \leq b, c < X_2 \leq d) = \int ...
Complex case
Complex random variable
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form P(Z \leq 1+2i) make no sense. However expressions of the form P(\Re \leq 1, \Im \leq 3) make sense. Therefore, we define the cumulative distribution of a complex random variables via the joint distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of their real and imaginary parts:
F_Z(z) = F_(\Re,\Im) = P(\Re \leq \Re , \Im \leq \Im).
Complex random vector
Generalization of yields
F_(\mathbf) = F_(\Re, \Im,\ldots,\Re, \Im) = \operatorname(\Re \leq \Re,\Im \leq \Im,\ldots,\Re \leq \Re,\Im \leq \Im)
as definition for the CDS of a complex random vector \mathbf = (Z_1,\ldots,Z_N)^T.
Use in statistical analysis
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. Cumulative frequency analysis
Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called ''frequency of non-exceedance ...
is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The empirical distribution function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function ...
is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various statistical hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
s. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from the same (unknown) population distribution.
Kolmogorov–Smirnov and Kuiper's tests
The Kolmogorov–Smirnov test
In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample wi ...
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 Kuiper's test is used in statistics to test that whether a given distribution, or family of distributions, is contradicted by evidence from a sample of data. It is named after Dutch mathematician Nicolaas Kuiper.
Kuiper's test is closely related t ...
is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used 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
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and a ...
* Distribution fitting
* Ogive (statistics)
In statistics, an ogive, also known as a cumulative frequency polygon, can refer to one of two things:
* any hand drawn graphic of a cumulative distribution function
In probability theory and statistics, the cumulative distribution functio ...
References
External links
*
{{DEFAULTSORT:Cumulative Distribution Function
Functions related to probability distributions