SkewNormalDistribution
SkewNormalDistribution
SkewNormalDistribution[μ,σ,α]
represents a skew-normal distribution with shape parameter α, location parameter μ, and scale parameter σ.
Details
- The probability density for value
in a skew-normal distribution is proportional to 
. - SkewNormalDistribution allows α and μ to be any real numbers and σ to be any positive real number.
- SkewNormalDistribution[α] is equivalent to SkewNormalDistribution[0,1,α].
- SkewNormalDistribution allows μ and σ to be any quantities of the same unit dimensions, and α to be a dimensionless quantity. »
- SkewNormalDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- SkewNormalDistribution[μ,σ,α] represents a continuous statistical distribution defined and supported over the set of real numbers and parametrized by a positive real number σ (a "scale parameter") and by two real numbers μ and α (a "location parameter" and a "shape parameter" respectively), which together determine the overall behavior of its probability density function (PDF). In general, the PDF of a skew-normal distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of α, μ, and σ. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of
. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) SkewNormalDistribution is a perhaps-skewed generalization of the normal distribution (NormalDistribution, sometimes referred to as the centralized normal distribution), and the one-parameter form SkewNormalDistribution[α] is equivalent to SkewNormalDistribution[0,1,α] (sometimes called the standard skew-normal distribution). - Aspects of the skew-normal distribution were first investigated by F. de Helguero in 1908 and Z. W. Birnbaum in 1950. Mathematically, the skew-normal distribution models both the largest component in a standardized binormal distribution (BinormalDistribution) and the maximum of two variates distributed according to the same normal distribution (NormalDistribution). The distribution is used somewhat frequently throughout statistics as well, emerging in the study of so-called threshold autoaggressive stochastic processes and in time series analysis. The skew-normal distribution can also be used to model a number of phenomena in fields such as physiology, finance, numerical and applied mathematics, telecommunications, and image analysis.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a skew-normal distribution. Distributed[x,SkewNormalDistribution[μ,σ,α]], written more concisely as xSkewNormalDistribution[μ,σ,α], can be used to assert that a random variable x is distributed according to a skew-normal distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for skew-normal distributions may be given using PDF[SkewNormalDistribution[μ,σ,α],x] and CDF[SkewNormalDistribution[μ,σ,α],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a skew-normal distribution, EstimatedDistribution to estimate a skew-normal parametric distribution from given data, and FindDistributionParameters to fit data to a skew-normal distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic skew-normal distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic skew-normal distribution.
- TransformedDistribution can be used to represent a transformed skew-normal distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a skew-normal distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving skew-normal distributions.
- SkewNormalDistribution is related to a number of other distributions. It is an immediate generalization of NormalDistribution, in the sense that the PDF of SkewNormalDistribution[μ,σ,0] is precisely that of NormalDistribution[μ,σ]. SkewNormalDistribution can also be realized as a transformation (TransformedDistribution) of both NormalDistribution and BinormalDistribution and is a limiting case of HalfNormalDistribution. SkewNormalDistribution satisfies a somewhat-novel identity, in that the PDF of SkewNormalDistribution[0,σ,α] is equivalent to 2PDF[NormalDistribution[0,σ],x] CDF[NormalDistribution[0,σ],α x]. SkewNormalDistribution is also related to NoncentralBetaDistribution, NoncentralFRatioDistribution, NoncentralChiSquareDistribution, LogNormalDistribution, and MultinormalDistribution.
Examples
open all close allBasic Examples (3)
Plot[Table[PDF[SkewNormalDistribution[0, 2, α], x], {α, {-3, 0.5, 4}}]//Evaluate, {x, -5, 5}, Filling -> Axis]Plot[Table[PDF[SkewNormalDistribution[0, σ, 2], x], {σ, {0.5, 1.5, 2}}]//Evaluate, {x, -2, 4}, PlotRange -> All, Filling -> Axis]PDF[SkewNormalDistribution[μ, σ, α], x]Cumulative distribution function:
Plot[Table[CDF[SkewNormalDistribution[α], x], {α, {-3, 0.5, 4}}]//Evaluate, {x, -5, 5}, Filling -> Axis]Plot[Table[CDF[SkewNormalDistribution[0, σ, 2], x], {σ, {0.5, 1.5, 2}}]//Evaluate, {x, -2, 4}, PlotRange -> All, Filling -> Axis]CDF[SkewNormalDistribution[μ, σ, α], x]Mean[SkewNormalDistribution[μ, σ, α]]Variance[SkewNormalDistribution[μ, σ, α]]Scope (8)
Generate a sample of pseudorandom numbers from a skew normal distribution:
data = RandomVariate[SkewNormalDistribution[0, 2, 3], 10 ^ 4];Compare its histogram to the PDF:
Show[
Histogram[data, 20, "PDF"],
Plot[PDF[SkewNormalDistribution[0, 2, 3], x], {x, -2, 7}, PlotStyle -> Thick]]Distribution parameters estimation:
sample = RandomVariate[SkewNormalDistribution[0, 3, -2], 10 ^ 3];Estimate the distribution parameters from sample data:
edist = EstimatedDistribution[sample, SkewNormalDistribution[μ, σ, α]]Compare a density histogram of the sample with the PDF of the estimated distribution:
Show[Histogram[sample, Automatic, "PDF"], Plot[PDF[edist, x], {x, -10, 4}, PlotStyle -> Thick]]Skewness depends only on the shape parameter α:
Skewness[SkewNormalDistribution[μ, σ, α]]Plot[%, {α, -10, 10}]Skewness is symmetric about the origin:
Skewness[SkewNormalDistribution[μ, σ, α]] + Skewness[SkewNormalDistribution[μ, σ, -α]]//SimplifyThe limiting value of skewness is finite and depends on the sign of α:
Limit[Skewness[SkewNormalDistribution[μ, σ, α]], α -> ∞]N[%]Kurtosis depends only on the shape parameter α:
Kurtosis[SkewNormalDistribution[μ, σ, α]]Plot[%, {α, -10, 10}]Kurtosis is symmetric about the origin and attains its minimum at 0:
Kurtosis[SkewNormalDistribution[μ, σ, 0]] /. α -> 0The limiting value is larger than kurtosis of NormalDistribution:
Limit[Kurtosis[SkewNormalDistribution[μ, σ, α]], α -> ∞]N[%]Different moments with closed forms as functions of parameters:
FormulaGrid[list_, type_] := Grid[...]FormulaGrid[Table[Moment[SkewNormalDistribution[μ, σ, α], k], {k, 3}], M]Closed form for symbolic order is known for the zero location parameter:
Moment[SkewNormalDistribution[0, σ, α], r]FormulaGrid[Table[CentralMoment[SkewNormalDistribution[μ, σ, α], k]//Factor, {k, 3}], CM]FormulaGrid[Table[FactorialMoment[SkewNormalDistribution[μ, σ, α], k]//FullSimplify, {k, 3}], FM]FormulaGrid[Table[Cumulant[SkewNormalDistribution[μ, σ, α], k]//FullSimplify, {k, 3}], C]Plot[Table[HazardFunction[SkewNormalDistribution[0, 2, α], x], {α, {-0.5, 1.5, 10}}]//Evaluate, {x, -3, 3}, Filling -> Axis]Plot[Table[HazardFunction[SkewNormalDistribution[0, σ, -0.2], x], {σ, {0.5, 1, 2}}]//Evaluate, {x, -2, 1}, Filling -> Axis]HazardFunction[SkewNormalDistribution[μ, σ, α], x]Plot[Table[Quantile[SkewNormalDistribution[0, 2, α], q], {α, {-3, -0.5, 8}}]//Evaluate, {q, 0, 1}, Filling -> Axis]Plot[Table[Quantile[SkewNormalDistribution[0, σ, 2], q], {σ, {1, 3, 8}}]//Evaluate, {q, 0, 1}, Filling -> Axis]Consistent use of Quantity in parameters yields QuantityDistribution:
arrivalTime𝒟 = SkewNormalDistribution[Quantity[3, "Hours"], Quantity[1.5, "Hours"], 1.5]Median[arrivalTime𝒟]Applications (3)
The height and weight for a group of people follow a binormal distribution with positive correlation of 0.6 and with means 180 cm and 90 kg, standard deviations 12 cm and 5 kg, respectively. The conditional probability of height for people who weigh more than 90 kg follows SkewNormalDistribution:
bn𝒟 = BinormalDistribution[{Quantity[180, "Centimeters"], Quantity[90, "Kilograms"]}, {Quantity[12, "Centimeters"], Quantity[5, "Kilograms"]}, ρ];pdf = D[CDF[bn𝒟, {Quantity[x, "Centimeters"], ∞}] - CDF[bn𝒟, {Quantity[x, "Centimeters"], Quantity[90, "Kilograms"]}], x] / Probability[Quantity[90, "Kilograms"] < y, {x, y}bn𝒟]//FullSimplify𝒮𝒩 = SkewNormalDistribution[Quantity[180, "Centimeters"], Quantity[12, "Centimeters"], (ρ/Sqrt[1 - ρ^2])];QuantityMagnitude[PDF[𝒮𝒩, Quantity[x, "Centimeters"]]] - pdf//FullSimplifyPlot the distribution density:
Plot[PDF[𝒮𝒩 /. ρ -> 0.6, Quantity[x, "Centemeters"]]//Evaluate, {x, 150, 220}, Filling -> Axis, AxesLabel -> {"cm"}]Compute four standard moments for this group:
Block[{ρ = 0.6}, {Mean[𝒮𝒩], StandardDeviation[𝒮𝒩], Skewness[𝒮𝒩], Kurtosis[𝒮𝒩]}]Compute the mean residual life function of a skew-normal random variate:
μ[α_, t_] = Expectation[xx > t, xSkewNormalDistribution[α], Assumptions -> α > 0] - tPlot the mean residual life function for several values of parameter 
, including the limiting case of normal variate, i.e. 
:
Plot[Table[Legended[μ[α, t], Row[{"α = ", α}]], {α, {0, 1 / 2, 2}}]//Evaluate, {t, -2, 5}]The finishing times for a 2010 Chicago marathon follow a SkewNormalDistribution:
𝒟 = SkewNormalDistribution[Quantity[234, "Minutes"], Quantity[74.37, "Minutes"], 1.54];Plot[PDF[𝒟, Quantity[x, "Minutes"]], {x, 100, 500}, Filling -> Axis, AxesLabel -> {"min"}]Mean[𝒟]UnitConvert[%, MixedUnit[{"Hours", "Minutes", "Seconds"}]]Find under what time half of the finishers crossed the finish line:
Median[𝒟]UnitConvert[%, MixedUnit[{"Hours", "Minutes", "Seconds"}]]The finishing time distribution is skewed to the right:
Skewness[𝒟]Properties & Relations (9)
Skew-normal distribution is closed under translation and scaling by a positive factor:
TransformedDistribution[k * u + loc, uSkewNormalDistribution[μ, σ, α]]Relationships to other distributions:
NormalDistribution is a special case of SkewNormalDistribution:
PDF[SkewNormalDistribution[μ, σ, 0], x]PDF[NormalDistribution[μ, σ], x]% - %%SkewNormalDistribution is a transformation of normal distribution:
TransformedDistribution[α Abs[u] + v, {u, v}ProductDistribution[{NormalDistribution[0, σ / Sqrt[1 + α ^ 2] ], 2}]]Probability density function can be expressed in terms of distribution functions of NormalDistribution:
2PDF[NormalDistribution[0, σ], x]CDF[NormalDistribution[0, σ], α x]PDF[SkewNormalDistribution[0, σ, α], x]% - %%//FullSimplifyHalfNormalDistribution is the limiting case of SkewNormalDistribution:
Limit[PDF[SkewNormalDistribution[0, Sqrt[π] / (Sqrt[2] θ), α], x], α -> ∞, Assumptions -> θ > 0 && x > 0]Refine[PDF[HalfNormalDistribution[θ], x], x > 0]% - %%
Limit[PDF[SkewNormalDistribution[0, Sqrt[π] / (Sqrt[2] θ), α], x], α -> ∞, Assumptions -> θ > 0 && x < 0]Refine[PDF[HalfNormalDistribution[θ], x], x < 0]SkewNormalDistribution is a transformation of BinormalDistribution:
𝒟 = TransformedDistribution[u Sign[v], {u, v}BinormalDistribution[ρ]];Block[{ρ = 0.7}, Show[Histogram[RandomVariate[𝒟, 10 ^ 4], 20, "PDF"], Plot[PDF[SkewNormalDistribution[0, 1, (ρ/Sqrt[1 - ρ^2])], x], {x, -3, 3}, PlotStyle -> Thick]]]The largest component of the standardized binormal distribution follows SkewNormalDistribution:
𝒟 = TransformedDistribution[Max[u, v], {u, v}BinormalDistribution[ρ]];Block[{ρ = 0.7}, Show[Histogram[RandomVariate[𝒟, 10 ^ 4], 20, "PDF"], Plot[PDF[SkewNormalDistribution[0, 1, Sqrt[(1 - ρ/1 + ρ)]], x], {x, -3, 3}, PlotStyle -> Thick]]]The maximum of two variates with the same NormalDistribution follows a SkewNormalDistribution:
PDF[OrderDistribution[{NormalDistribution[μ, σ], 2}, 2], x]PDF[SkewNormalDistribution[μ, σ, 1], x]% - %%//SimplifyNeat Examples (1)
PDFs for different α values with CDF contours:
cdf = Function[{x, α}, Evaluate[CDF[SkewNormalDistribution[0, 2, α], x]]];
ql = {0.025, 0.10, 0.25, 0.5, 0.75, 0.90, 0.975};
cl = Table[ColorData["Rainbow"][q], {q, Join[{0.0}, ql]}];Legended[Plot3D[PDF[SkewNormalDistribution[0, 2, α], x], {x, -3, 3}, {α, -3, 3}, PlotTheme -> "Marketing", MeshFunctions -> {cdf}, Mesh -> {ql}, MeshStyle -> GrayLevel[0.8], MeshShading -> cl, AxesLabel -> Automatic, PlotPoints -> 80, BaseStyle -> Opacity[0.9], ImageSize -> 400], BarLegend["Rainbow", ql, LegendLabel -> "prob"]]
Wolfram Research (2010), SkewNormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SkewNormalDistribution.html (updated 2016).
Text
Wolfram Research (2010), SkewNormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SkewNormalDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "SkewNormalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/SkewNormalDistribution.html.
APA
Wolfram Language. (2010). SkewNormalDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SkewNormalDistribution.html