CDF
CDF[dist,x]
gives the cumulative distribution function for the distribution dist evaluated at x.
CDF[dist,{x1,x2,…}]
gives the multivariate cumulative distribution function for the distribution dist evaluated at {x1,x2,…}.
CDF[dist]
gives the CDF as a pure function.
Details
- CDF[dist,x] gives the probability that an observed value will be less than or equal to x.
- CDF[dist,x] is equivalent to Probability[ξ≤x,ξdist].
- CDF[dist,{x1,…,xn}] is equivalent to Probability[ξ1≤x1∧⋯∧ξn≤xn,{ξ1,…,ξn}dist].
- CDF[dist,x] is equivalent to 1-SurvivalFunction[dist,x].
Examples
open allclose allBasic Examples (4)
Scope (21)
Parametric Distributions (4)
Nonparametric Distributions (4)
CDF for nonparametric distributions:
Plot the CDF for a histogram distribution:
Closed-form expression for the CDF of a kernel mixture distribution:
Derived Distributions (10)
Product of independent distributions:
Component mixture distribution:
Quadratic transformation of a discrete distribution:
Parameter mixture distribution:
Formula distribution defined by its PDF:
Defined by its SurvivalFunction:
The CDF for QuantityDistribution assumes the argument is a Quantity with compatible units:
Random Processes (3)
Find the CDF for a SliceDistribution of a discrete-state random process:
A continuous-state random process:
Find the multiple time-slice CDF for a discrete-state process:
A multi-slice for a continuous-state process:
Find the CDF for the StationaryDistribution of a discrete-state random process:
Generalizations & Extensions (1)
CDF threads element-wise over lists:
Applications (5)
Plot the CDF for a standard normal distribution:
Plot the CDF for a binomial distribution:
Compute the probability of 
for a 
distribution with 20 degrees of freedom:
Compute the probability of 
for the same distribution:
Perform a probability integral transform on data by mapping the CDF over it:
The transformed data is uniformly distributed if the original data came from the chosen distribution:
Comparing transformed data to a uniform distribution and comparing original data to original distribution should give identical results for all applicable tests:
Define a general survival distribution function (SDF) as used in actuarial science:
Compare with the expression given by SurvivalFunction:
Define the force of mortality (FM):
Compare with the expression given by HazardFunction:
Properties & Relations (12)
The probability of 
for a univariate distribution is given by its CDF:
The probability of 
for a multivariate distribution is given by its CDF:
A univariate CDF is 0 at 
and 1 at 
:
A multivariate CDF has value 0 at 
and 1 at 
:
The CDF is the integral of the PDF for continuous distributions 
:
The CDF is the sum of the PDF for discrete distributions 
:
CDF and InverseCDF are inverses for continuous distributions:
Compositions of CDF and InverseCDF give step functions for a discrete distribution:
CDF and Quantile are inverses for continuous distributions:
The sum of the CDF and the survival function is 1:
ProbabilityPlot generates a parametric plot of the empirical CDF vs estimated CDF:
CDF is a right-continuous function with left limits:
Possible Issues (2)
Symbolic closed forms do not exist for some distributions:
Substitution of invalid values into symbolic formulas can give results that are not meaningful:
When CDF is given an explicit value as an argument, it does complete checking and does not produce invalid results:
Neat Examples (1)
CDF for a bivariate censored distribution:
Text
Wolfram Research (2007), CDF, Wolfram Language function, https://reference.wolfram.com/language/ref/CDF.html (updated 2010).
CMS
Wolfram Language. 2007. "CDF." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010. https://reference.wolfram.com/language/ref/CDF.html.
APA
Wolfram Language. (2007). CDF. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CDF.html