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Wolfram Language & System Documentation Center
SurvivalFunction
  • See Also
    • CDF
    • InverseSurvivalFunction
    • HazardFunction
    • RarerProbability
    • PDF
    • Quantile
    • Probability
    • NProbability
    • Integrate
    • Sum
    • Histogram
    • Histogram3D
  • Related Guides
    • Statistical Distribution Functions
    • Reliability
    • Random Variables
    • System Model Analytics & Design
    • See Also
      • CDF
      • InverseSurvivalFunction
      • HazardFunction
      • RarerProbability
      • PDF
      • Quantile
      • Probability
      • NProbability
      • Integrate
      • Sum
      • Histogram
      • Histogram3D
    • Related Guides
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      • Reliability
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      • System Model Analytics & Design

SurvivalFunction[dist,x]

gives the survival function for the distribution dist evaluated at x.

SurvivalFunction[dist,{x1,x2,…}]

gives the multivariate survival function for the distribution dist evaluated at {x1,x2,…}.

SurvivalFunction[dist]

gives the survival function as a pure function.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Parametric Distributions  
Nonparametric Distributions  
Derived Distributions  
Random Processes  
Generalizations & Extensions  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
Cite this Page
BUILT-IN SYMBOL
  • See Also
    • CDF
    • InverseSurvivalFunction
    • HazardFunction
    • RarerProbability
    • PDF
    • Quantile
    • Probability
    • NProbability
    • Integrate
    • Sum
    • Histogram
    • Histogram3D
  • Related Guides
    • Statistical Distribution Functions
    • Reliability
    • Random Variables
    • System Model Analytics & Design
    • See Also
      • CDF
      • InverseSurvivalFunction
      • HazardFunction
      • RarerProbability
      • PDF
      • Quantile
      • Probability
      • NProbability
      • Integrate
      • Sum
      • Histogram
      • Histogram3D
    • Related Guides
      • Statistical Distribution Functions
      • Reliability
      • Random Variables
      • System Model Analytics & Design

SurvivalFunction

SurvivalFunction[dist,x]

gives the survival function for the distribution dist evaluated at x.

SurvivalFunction[dist,{x1,x2,…}]

gives the multivariate survival function for the distribution dist evaluated at {x1,x2,…}.

SurvivalFunction[dist]

gives the survival function as a pure function.

Details

  • SurvivalFunction is also known as a complementary cumulative distribution function or a reliability function.
  • SurvivalFunction[dist,x] gives the probability that an observed value is greater than x.
  • SurvivalFunction[dist,x] is equivalent to Probability[ξ>x,ξ∈dist].
  • SurvivalFunction[dist,{x1,…,xn}] is equivalent to Probability[ξ1>x1∧⋯∧ξn>xn,{ξ1,…,ξn}dist].
  • SurvivalFunction[dist,x] is equivalent to 1-CDF[dist,x].

Examples

open all close all

Basic Examples  (4)

A survival function for a continuous univariate distribution:

A survival function for a discrete univariate distribution:

A survival function for a continuous multivariate distribution:

A survival function for a discrete multivariate distribution:

Scope  (25)

Parametric Distributions  (8)

Obtain exact numeric results:

Obtain a machine-precision result:

Obtain a result at any precision for a continuous distribution:

Obtain a result at any precision for a discrete distribution with inexact parameters:

Survival function for a multivariate distribution:

Obtain a symbolic expression for the survival function:

Obtain pure function result:

Evaluate at a value:

SurvivalFunction threads elementwise over lists:

Multivariate distributions:

Nonparametric Distributions  (4)

Survival function for nonparametric distributions:

Compare with the value for the underlying parametric distribution:

Plot the survival function for a histogram distribution:

Closed form expression for the survival function of a kernel mixture distribution:

Plot of the survival function of a bivariate smooth kernel distribution:

Derived Distributions  (10)

Product of independent distributions:

Component mixture distribution:

Quadratic transformation of a discrete distribution:

Censored distribution:

Compare survival function of the censored distribution with the original:

Truncated distribution:

Compare survival function of the truncated distribution with the original:

Parameter mixture distribution:

Copula distribution:

Formula distributions defined by its PDF:

Defined by its CDF:

Defined by its survival function:

Marginal distribution:

The survival function for QuantityDistribution assumes the argument is a Quantity with compatible units:

This allows for direct quantity substitution:

Compare with the direct use of the quantity argument:

Random Processes  (3)

Find the survival function for a SliceDistribution of a discrete-state random process:

A continuous-state random process:

Find the multiple time-slice survival function for a discrete-state process:

A multi-slice for a continuous-state process:

Survival function for the StationaryDistribution of a discrete-state random process:

Generalizations & Extensions  (1)

SurvivalFunction threads element-wise over lists:

Multivariate distributions:

Applications  (2)

Compute the probability of for a distribution with 20 degrees of freedom:

Compute the probability of for the same distribution:

Probability of getting at least one six in 6 throws of a regular six‐sided die:

Probability of getting at least two sixes in 12 throws:

Probability of getting at least three sixes in 18 throws:

Getting at least one six in 6 throws is the most favorable bet:

Properties & Relations  (6)

The probability of for a continuous univariate distribution is given by SurvivalFunction:

The survival function has value 1 at and is 0 at :

The sum of the survival function and the CDF is 1:

SurvivalFunction and InverseSurvivalFunction are inverses for continuous distributions:

Compositions of SurvivalFunction and InverseSurvivalFunction give step functions for a discrete distribution:

Calculate the PDF of a continuous univariate distribution:

Possible Issues  (2)

Symbolic closed forms do not exist for some distributions:

Numerical evaluation works:

Substitution of invalid values into symbolic outputs gives results that are not meaningful:

Passing it as an argument, it stays unevaluated:

See Also

CDF  InverseSurvivalFunction  HazardFunction  RarerProbability  PDF  Quantile  Probability  NProbability  Integrate  Sum  Histogram  Histogram3D

Function Repository: MortalityConvert

Related Guides

    ▪
  • Statistical Distribution Functions
  • ▪
  • Reliability
  • ▪
  • Random Variables
  • ▪
  • System Model Analytics & Design

History

Introduced in 2010 (8.0)

Wolfram Research (2010), SurvivalFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/SurvivalFunction.html.

Text

Wolfram Research (2010), SurvivalFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/SurvivalFunction.html.

CMS

Wolfram Language. 2010. "SurvivalFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SurvivalFunction.html.

APA

Wolfram Language. (2010). SurvivalFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SurvivalFunction.html

BibTeX

@misc{reference.wolfram_2025_survivalfunction, author="Wolfram Research", title="{SurvivalFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/SurvivalFunction.html}", note=[Accessed: 04-February-2026]}

BibLaTeX

@online{reference.wolfram_2025_survivalfunction, organization={Wolfram Research}, title={SurvivalFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/SurvivalFunction.html}, note=[Accessed: 04-February-2026]}

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