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Cumulative distribution function
From Wikipedia, the free encyclopedia This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2010)
Cumulative distribution function for the normal distribution
In probability theory and statistics, the cumulative distribution function CDF!, or "ust distribution
function, describes the probability that a real#valued random variable X $ith a given probability distribution $ill be found to have a value less than or e%ual to x. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.
Contents
• & 'efinition
• ( Properties
• ) *+amples
o .( Folded cumulative distribution
• - ultivariate case
• 4 2ee also
Definition
The cumulative distribution function of a real#valued random variable X is the function given by
$here the right#hand side represents the probability that the random variable X takes on a value less than or e%ual to x. The probability that X lies in the semi#closed interval a, b8, $here a 9 b, is therefore
In the definition above, the :less than or e%ual to: sign, :;:, is a convention, not a universally used one e.g. <ungarian literature uses :9:!, but is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. oreover, important formulas like Paul =>vy3s inversion formula for the characteristic function also rely on the :less than or e%ual: formulation.
If treating several random variables X , Y , ... etc. the corresponding letters are used as subscripts $hile, if treating only one, the subscript is usually omitted. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lo$er#case f used for probability density functions and probability mass functions. This applies $hen discussing general distributions? some specific distributions have their o$n conventional notation, for e+ample the normal distribution.
The C'F of a continuous random variable X can be e+pressed as the integral of its probability density function @A as follo$s?
In the case of a random variable X $hich has distribution having a discrete component at a value b,
If F X is continuous at b, this e%uals Bero and there is no discrete component at b.
Properties
*very cumulative distribution function F is non#decreasing and right#continuous, $hich makes it a cdlg function. Furthermore,
*very function $ith these four properties is a C'F, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable.
If X is a purely discrete random variable, then it attains values x&, x(, ... $ith probability pi D P xi!, and the C'F of X $ill be discontinuous at the points xi and constant in bet$een?
If the C'F F of a real valued random variable X is continuous, then X is a continuous random variableE if furthermore F is absolutely continuous, then there e+ists a =ebesgue#integrable function f x! such that
for all real numbers a and b. The function f is e%ual to the derivative of F almost every$here, and it is called the probability density function of the distribution of X .
Examples
is uniformly distributed on the unit interval GH, &8. Then the C'F of is given by
Complementary cumulative distribution function (tail distribution)
2ometimes, it is useful to study the opposite %uestion and ask ho$ 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
This has applications in statistical hypothesis testing, for e+ample, because the one#sided p#value is the probability of observing a test statistic at least as e+treme as the one observed. Thus, provided that the test statistic, T , has a continuous distribution, the one#sided p#value is simply given by the ccdf? for an observed value t of the test statistic
In survival analysis,
is called the survival function and denoted , $hile the term reliability
function is common in engineering.
Properties
• For a non#negative continuous random variable having an e+pectation, arkov3s ine%uality states thatG&8
• s , and in fact provided that is finite.
Proof?Gcitation needed
8 ssuming A has a density function f, for any
Then, on recogniBing and rearranging terms,
as claimed.
Folded cumulative distribution
*+ample of the folded cumulative distribution for a normal distribution function $ith an e+pected value of H and a standard deviation of &.
While the plot of a cumulative distribution often has an 2#like shape, an alternative illustration is the folded
cumulative distribution or mountain plot, $hich folds the top half of the graph over ,G(8G)8 thus using t$o scales, one for the upslope and another for the do$nslope. This form of illustration emphasises the median and dispersion the mean absolute deviation from the medianG8! of the distribution or of the empirical results.
Inverse distribution function (quantile function)
If the C'F F is strictly increasing and continuous then is the uni%ue real number
such that . In such a case, this defines the inverse distribution function or %uantile function.
Unfortunately, the distribution does not, in general, have an inverse. ne may define, for , the generalized inverse distribution function? Gclarification needed 8
• *+ample &? The median is .
The inverse of the cdf is called the %uantile function.
&.
. if and only if
-. If has a distribution then is distributed as . This is used in random number generation using the inverse transform sampling#method.
/. If is a collection of independent #distributed random variables defined on the same sample
space, then there e+ist random variables such that
ultivariate case
When dealing simultaneously $ith more than one random variable the joint cumulative distribution function can also be defined. For e+ample, for a pair of random variables XY , the "oint C'F is given by
$here the right#hand side represents the probability that the random variable X takes on a value less than or e%ual to x and that Y takes on a value less than or e%ual to y.
*very multivariate C'F is?
(. 6ight#continuous for each of its variables.
).
!se in statistical analysis
The concept of the cumulative distribution function makes an e+plicit appearance in statistical analysis in t$o similar! $ays. Cumulative fre%uency analysis is the analysis of the fre%uency of occurrence of values of a phenomenon less than a reference value. The empirical distribution function is a formal direct estimate of the cumulative distribution function for $hich simple statistical properties can be derived and $hich can form the basis of various statistical hypothesis tests. 2uch tests can assess $hether there is evidence against a sample of data having arisen from a given distribution, or evidence against t$o samples of data having arisen from the same unkno$n! population distribution.
"olmogorov#$mirnov and "uiper%s tests
The 0olmogorov12mirnov test is based on cumulative distribution functions and can be used to test to see $hether t$o empirical distributions are different or $hether an empirical distribution is different from an ideal distribution. The closely related 0uiper3s test is useful if the domain of the distribution is cyclic as in day of the $eek. For instance 0uiper3s 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 $eek or day of the month.
$ee also
• 'escriptive statistics
• 'istribution fitting
J$illinger, 'anielE 0okoska, 2tephen (H&H!. !"! #tandard $robability and #tatistics Tables and
For%ulae. C6C Press. p. 7. I2KL 745#&#-555#H-7#(. Mentle, N.*. (HH7!. !o%putational #tatistics. 2pringer . I2KL 745#H#)54#75&-#&. 6etrieved (H&H#
H5#H/.G pa&e needed 8 onti, 0.=. &77-!. :Folded *mpirical 'istribution Function Curves ountain Plots!:. The
'%erican #tatistician '? )(1)-. N2T6 (/5-4H.
&. Aue, N. <.E Titterington, '. . (H&&!. :The p#folded cumulative distribution function and
the mean absolute deviation from the p#%uantile:. #tatistics $robability etters * 5!? &&471&&5(. doi?&H.&H&/O".spl.(H&&.H).H&.9
External lin+s
*dit links
• This page $as last modified on &5 Lovember (H&, at &&?-H.
• Te+t is available under the Creative Commons ttribution#2harelike =icenseE additional terms may apply. Ky using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non#profit organiBation.
From Wikipedia, the free encyclopedia This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2010)
Cumulative distribution function for the normal distribution
In probability theory and statistics, the cumulative distribution function CDF!, or "ust distribution
function, describes the probability that a real#valued random variable X $ith a given probability distribution $ill be found to have a value less than or e%ual to x. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.
Contents
• & 'efinition
• ( Properties
• ) *+amples
o .( Folded cumulative distribution
• - ultivariate case
• 4 2ee also
Definition
The cumulative distribution function of a real#valued random variable X is the function given by
$here the right#hand side represents the probability that the random variable X takes on a value less than or e%ual to x. The probability that X lies in the semi#closed interval a, b8, $here a 9 b, is therefore
In the definition above, the :less than or e%ual to: sign, :;:, is a convention, not a universally used one e.g. <ungarian literature uses :9:!, but is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. oreover, important formulas like Paul =>vy3s inversion formula for the characteristic function also rely on the :less than or e%ual: formulation.
If treating several random variables X , Y , ... etc. the corresponding letters are used as subscripts $hile, if treating only one, the subscript is usually omitted. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lo$er#case f used for probability density functions and probability mass functions. This applies $hen discussing general distributions? some specific distributions have their o$n conventional notation, for e+ample the normal distribution.
The C'F of a continuous random variable X can be e+pressed as the integral of its probability density function @A as follo$s?
In the case of a random variable X $hich has distribution having a discrete component at a value b,
If F X is continuous at b, this e%uals Bero and there is no discrete component at b.
Properties
*very cumulative distribution function F is non#decreasing and right#continuous, $hich makes it a cdlg function. Furthermore,
*very function $ith these four properties is a C'F, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable.
If X is a purely discrete random variable, then it attains values x&, x(, ... $ith probability pi D P xi!, and the C'F of X $ill be discontinuous at the points xi and constant in bet$een?
If the C'F F of a real valued random variable X is continuous, then X is a continuous random variableE if furthermore F is absolutely continuous, then there e+ists a =ebesgue#integrable function f x! such that
for all real numbers a and b. The function f is e%ual to the derivative of F almost every$here, and it is called the probability density function of the distribution of X .
Examples
is uniformly distributed on the unit interval GH, &8. Then the C'F of is given by
Complementary cumulative distribution function (tail distribution)
2ometimes, it is useful to study the opposite %uestion and ask ho$ 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
This has applications in statistical hypothesis testing, for e+ample, because the one#sided p#value is the probability of observing a test statistic at least as e+treme as the one observed. Thus, provided that the test statistic, T , has a continuous distribution, the one#sided p#value is simply given by the ccdf? for an observed value t of the test statistic
In survival analysis,
is called the survival function and denoted , $hile the term reliability
function is common in engineering.
Properties
• For a non#negative continuous random variable having an e+pectation, arkov3s ine%uality states thatG&8
• s , and in fact provided that is finite.
Proof?Gcitation needed
8 ssuming A has a density function f, for any
Then, on recogniBing and rearranging terms,
as claimed.
Folded cumulative distribution
*+ample of the folded cumulative distribution for a normal distribution function $ith an e+pected value of H and a standard deviation of &.
While the plot of a cumulative distribution often has an 2#like shape, an alternative illustration is the folded
cumulative distribution or mountain plot, $hich folds the top half of the graph over ,G(8G)8 thus using t$o scales, one for the upslope and another for the do$nslope. This form of illustration emphasises the median and dispersion the mean absolute deviation from the medianG8! of the distribution or of the empirical results.
Inverse distribution function (quantile function)
If the C'F F is strictly increasing and continuous then is the uni%ue real number
such that . In such a case, this defines the inverse distribution function or %uantile function.
Unfortunately, the distribution does not, in general, have an inverse. ne may define, for , the generalized inverse distribution function? Gclarification needed 8
• *+ample &? The median is .
The inverse of the cdf is called the %uantile function.
&.
. if and only if
-. If has a distribution then is distributed as . This is used in random number generation using the inverse transform sampling#method.
/. If is a collection of independent #distributed random variables defined on the same sample
space, then there e+ist random variables such that
ultivariate case
When dealing simultaneously $ith more than one random variable the joint cumulative distribution function can also be defined. For e+ample, for a pair of random variables XY , the "oint C'F is given by
$here the right#hand side represents the probability that the random variable X takes on a value less than or e%ual to x and that Y takes on a value less than or e%ual to y.
*very multivariate C'F is?
(. 6ight#continuous for each of its variables.
).
!se in statistical analysis
The concept of the cumulative distribution function makes an e+plicit appearance in statistical analysis in t$o similar! $ays. Cumulative fre%uency analysis is the analysis of the fre%uency of occurrence of values of a phenomenon less than a reference value. The empirical distribution function is a formal direct estimate of the cumulative distribution function for $hich simple statistical properties can be derived and $hich can form the basis of various statistical hypothesis tests. 2uch tests can assess $hether there is evidence against a sample of data having arisen from a given distribution, or evidence against t$o samples of data having arisen from the same unkno$n! population distribution.
"olmogorov#$mirnov and "uiper%s tests
The 0olmogorov12mirnov test is based on cumulative distribution functions and can be used to test to see $hether t$o empirical distributions are different or $hether an empirical distribution is different from an ideal distribution. The closely related 0uiper3s test is useful if the domain of the distribution is cyclic as in day of the $eek. For instance 0uiper3s 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 $eek or day of the month.
$ee also
• 'escriptive statistics
• 'istribution fitting
J$illinger, 'anielE 0okoska, 2tephen (H&H!. !"! #tandard $robability and #tatistics Tables and
For%ulae. C6C Press. p. 7. I2KL 745#&#-555#H-7#(. Mentle, N.*. (HH7!. !o%putational #tatistics. 2pringer . I2KL 745#H#)54#75&-#&. 6etrieved (H&H#
H5#H/.G pa&e needed 8 onti, 0.=. &77-!. :Folded *mpirical 'istribution Function Curves ountain Plots!:. The
'%erican #tatistician '? )(1)-. N2T6 (/5-4H.
&. Aue, N. <.E Titterington, '. . (H&&!. :The p#folded cumulative distribution function and
the mean absolute deviation from the p#%uantile:. #tatistics $robability etters * 5!? &&471&&5(. doi?&H.&H&/O".spl.(H&&.H).H&.9
External lin+s
*dit links
• This page $as last modified on &5 Lovember (H&, at &&?-H.
• Te+t is available under the Creative Commons ttribution#2harelike =icenseE additional terms may apply. Ky using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non#profit organiBation.