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 allBasic Examples (4)
A survival function for a continuous univariate distribution:
SurvivalFunction[NormalDistribution[0, 1], x]Plot[%, {x, -3, 3}, Filling -> Axis]A survival function for a discrete univariate distribution:
SurvivalFunction[GeometricDistribution[1 / 3], x]DiscretePlot[%, {x, -1, 10}, ExtentSize -> Right, ExtentMarkers -> {"Filled", "Empty"}]A survival function for a continuous multivariate distribution:
Plot3D[SurvivalFunction[BinormalDistribution[1 / 2], {x, y}], {x, -3, 3}, {y, -3, 3}, PlotRange -> {0, 1}, MaxRecursion -> 0]A survival function for a discrete multivariate distribution:
DiscretePlot3D[SurvivalFunction[MultinomialDistribution[10, {1 / 3, 2 / 3}], {x, y}], {x, -1, 7}, {y, -1, 7}, ExtentSize -> Right]Scope (25)
Parametric Distributions (8)
SurvivalFunction[WeibullDistribution[2, 5], 4]SurvivalFunction[NegativeBinomialDistribution[20, 1 / 3], 5]Obtain a machine-precision result:
SurvivalFunction[WeibullDistribution[2, 5], 4.]Obtain a result at any precision for a continuous distribution:
SurvivalFunction[WeibullDistribution[2, 5], N[4, 25]]Obtain a result at any precision for a discrete distribution with inexact parameters:
SurvivalFunction[NegativeBinomialDistribution[20, N[1 / 3, 30]], 5]Survival function for a multivariate distribution:
SurvivalFunction[DirichletDistribution[{2, 4, 5}], {1 / 5, 2 / 3}]Obtain a symbolic expression for the survival function:
SurvivalFunction[ChiSquareDistribution[ν], x]SurvivalFunction[UniformDistribution[{{a, b}, {c, d}}], {x, y}]SurvivalFunction[GammaDistribution[1, 2]]%[3]SurvivalFunction threads elementwise over lists:
SurvivalFunction[NormalDistribution[], {0.0, 0.2, 0.3}]SurvivalFunction[BinormalDistribution[1 / 2], {{0.0, 0.0}, {0.2, 0.2}, {0.3, 0.3}}]Nonparametric Distributions (4)
Survival function for nonparametric distributions:
r = RandomVariate[NormalDistribution[], 10 ^ 4];SurvivalFunction[HistogramDistribution[r], 0.2]SurvivalFunction[SmoothKernelDistribution[r], 0.2]SurvivalFunction[KernelMixtureDistribution[r], 0.2]Compare with the value for the underlying parametric distribution:
SurvivalFunction[NormalDistribution[], 0.2]Plot the survival function for a histogram distribution:
Plot[SurvivalFunction[HistogramDistribution[RandomVariate[NormalDistribution[], 10 ^ 3]], x]//Evaluate, {x, -3, 3}, Filling -> Axis, Exclusions -> None]Closed form expression for the survival function of a kernel mixture distribution:
SurvivalFunction[KernelMixtureDistribution[RandomVariate[GammaDistribution[1, 2], 10
]], x]Plot of the survival function of a bivariate smooth kernel distribution:
Plot3D[SurvivalFunction[SmoothKernelDistribution[RandomVariate[BinormalDistribution[1 / 3], 30]], {x, y}]//Evaluate, {x, -4, 4}, {y, -4, 4}, PlotRange -> {0, 1.2}]Derived Distributions (10)
Product of independent distributions:
𝒟 = ProductDistribution[TriangularDistribution[{2, 4}], TriangularDistribution[{1, 7}]];sf = SurvivalFunction[𝒟, {x, y}]Plot3D[sf, {x, 1, 5}, {y, 0, 8}, PlotRange -> All, Exclusions -> None]Component mixture distribution:
𝒟 = MixtureDistribution[{1, 4}, {NormalDistribution[a, b], NormalDistribution[c, d]}];sf = SurvivalFunction[𝒟, x]Plot[sf /. {a -> 0, b -> 1, c -> 6, d -> 3 / 2}, {x, -2, 10}, Filling -> Axis]Quadratic transformation of a discrete distribution:
𝒟 = TransformedDistribution[x ^ 2, xPoissonDistribution[2]];SurvivalFunction[𝒟, y]DiscretePlot[%, {y, 0, 50}, PlotRange -> {0, 0.7}]𝒟 = CauchyDistribution[0, 1];
𝒞 = CensoredDistribution[{-2, 4}, 𝒟];SurvivalFunction[𝒞, x]Compare survival function of the censored distribution with the original:
Plot[{SurvivalFunction[𝒟, x], %}, {x, -3, 6}, Filling -> Axis, PlotLegends -> {"𝒟", "𝒞"}]𝒟 = TriangularDistribution[{1, 4}];
𝒯 = TruncatedDistribution[{2, 3}, 𝒟];SurvivalFunction[𝒯, x]Compare survival function of the truncated distribution with the original:
Plot[{SurvivalFunction[𝒟, x], %}, {x, 1, 4}, Filling -> Axis, Exclusions -> None, PlotLegends -> {"𝒟", "𝒯"}]Parameter mixture distribution:
𝒟 = ParameterMixtureDistribution[GeometricDistribution[r], rUniformDistribution[{1 / 2, 2 / 3}]];SurvivalFunction[𝒟, x]Plot[%, {x, -2, 4}, Filling -> Axis]𝒟 = CopulaDistribution[{"AMH", 1 / 3}, {UniformDistribution[{2, 4}], UniformDistribution[{3, 7}]}];sf = SurvivalFunction[𝒟, {x, y}]//SimplifyPlot3D[sf, {x, 0, 5}, {y, 0, 10}]Formula distributions defined by its PDF:
𝒟 = ProbabilityDistribution[(Sqrt[2] / Pi)(1 / (1 + x ^ 4)), {x, -Infinity, Infinity}];SurvivalFunction[𝒟, x]𝒟 = ProbabilityDistribution[{"CDF", Piecewise[{{-2 + x, 2 ≤ x ≤ 3}, {1, x > 3}}, 0]}, {x, -Infinity, Infinity}];SurvivalFunction[𝒟, x]Defined by its survival function:
𝒟 = ProbabilityDistribution[{"SF", Piecewise[{{1 - 2(-2 + x) ^ 2, 2 ≤ x ≤ 5 / 2}, {2(3 - x) ^ 2, 5 / 2 < x ≤ 3}, {1, x < 2}}, 0]}, {x, -Infinity, Infinity}];SurvivalFunction[𝒟, x]𝒟 = MarginalDistribution[ProbabilityDistribution[E^-(y^2/2) π^-3 / 2 (1 + x^4)^-1, {x, -∞, ∞}, {y, -∞, ∞}], 2];SurvivalFunction[𝒟, y]The survival function for QuantityDistribution assumes the argument is a Quantity with compatible units:
𝒟 = NormalDistribution[Quantity[4, "Meters"], Quantity[1 / 2, "Meters"]]SurvivalFunction[𝒟, x]This allows for direct quantity substitution:
% /. x -> Quantity[387., "Centimeters"]Compare with the direct use of the quantity argument:
SurvivalFunction[𝒟, Quantity[387., "Centimeters"]]Random Processes (3)
Find the survival function for a SliceDistribution of a discrete-state random process:
SurvivalFunction[PoissonProcess[μ][2], x]DiscretePlot[Evaluate[% /. μ -> 2], {x, -5, 10}, ExtentSize -> 0.5]A continuous-state random process:
SurvivalFunction[WienerProcess[][2], x]Plot[%, {x, -5, 3}, Filling -> Axis]Find the multiple time-slice survival function for a discrete-state process:
SurvivalFunction[PoissonProcess[μ][{2, 3}], {x, y}]DiscretePlot3D[Evaluate[% /. μ -> 2], {x, -5, 6}, {y, -5, 6}, ExtentSize -> 0.5]A multi-slice for a continuous-state process:
Plot3D[SurvivalFunction[WienerProcess[][{2, 3}], {x, y}], {x, -5, 3}, {y, -5, 3}]Survival function for the StationaryDistribution of a discrete-state random process:
SurvivalFunction[StationaryDistribution[QueueingProcess[λ, μ, 2]], x]//FullSimplifyDiscretePlot[Evaluate[% /. {μ -> 6, λ -> 5.9}], {x, -3, 8}, ExtentSize -> 0.5]Generalizations & Extensions (1)
SurvivalFunction threads element-wise over lists:
SurvivalFunction[NormalDistribution[], {0.2, 0.3}]{SurvivalFunction[NormalDistribution[], 0.2], SurvivalFunction[NormalDistribution[], 0.3]}SurvivalFunction[BinormalDistribution[1 / 2], {{0.0, 0.0}, {0.2, 0.2}, {0.3, 0.3}}]Applications (2)
Compute the probability of 
for a 
distribution with 20 degrees of freedom:
SurvivalFunction[StudentTDistribution[20], 3.]NProbability[t > 3, tStudentTDistribution[20]]Compute the probability of 
for the same distribution:
1 - SurvivalFunction[StudentTDistribution[20], 3.5]NProbability[t < 3.5, tStudentTDistribution[20]]Probability of getting at least one six in 6 throws of a regular six‐sided die:
pA = SurvivalFunction[BinomialDistribution[6, 1 / 6], 0]Probability of getting at least two sixes in 12 throws:
pB = SurvivalFunction[BinomialDistribution[12, 1 / 6], 1]Probability of getting at least three sixes in 18 throws:
pC = SurvivalFunction[BinomialDistribution[18, 1 / 6], 2]Getting at least one six in 6 throws is the most favorable bet:
pA == Max[pA, pB, pC]DiscretePlot[SurvivalFunction[BinomialDistribution[6 n, 1 / 6], n - 1], {n, 1, 12}, AxesOrigin -> {0, 1 / 2}]Properties & Relations (6)
The probability of 
for a continuous univariate distribution is given by SurvivalFunction:
Probability[x > a, xNormalDistribution[]]SurvivalFunction[NormalDistribution[], a]The survival function has value 1 at 
and is 0 at 
:
SurvivalFunction[NormalDistribution[], -Infinity]SurvivalFunction[NormalDistribution[], Infinity]The sum of the survival function and the CDF is 1:
SurvivalFunction[UniformDistribution[], x]CDF[UniformDistribution[], x]% + %%//SimplifySurvivalFunction and InverseSurvivalFunction are inverses for continuous distributions:
dist = ExponentialDistribution[λ];Assuming[x > 0 && λ > 0, Simplify[InverseSurvivalFunction[dist, SurvivalFunction[dist, x]] == x]]Assuming[1 > q > 0 && λ > 0, Simplify[SurvivalFunction[dist, InverseSurvivalFunction[dist, q]] == q]]Compositions of SurvivalFunction and InverseSurvivalFunction give step functions for a discrete distribution:
dist = PoissonDistribution[2];Plot[SurvivalFunction[dist, InverseSurvivalFunction[dist, q]], {q, 0, 1}]Plot[InverseSurvivalFunction[dist, SurvivalFunction[dist, y]], {y, 0, 20}]Calculate the PDF of a continuous univariate distribution:
dist = NormalDistribution[μ, σ];PDF[dist, x] == -D[SurvivalFunction[dist, x], x]Possible Issues (2)
Symbolic closed forms do not exist for some distributions:
SurvivalFunction[StableDistribution[0, 1.8, -0.5, 1, 2], x]SurvivalFunction[StableDistribution[0, 1.8, -0.5, 1, 2], 0.3]Substitution of invalid values into symbolic outputs gives results that are not meaningful:
SurvivalFunction[CauchyDistribution[2, 3], y] /. {y -> 1. + I}Passing it as an argument, it stays unevaluated:
SurvivalFunction[CauchyDistribution[2, 3], 1. + I]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