SkewNormalDistribution

Generate a set of pseudorandom numbers that are skew normally distributed:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness depends only on the shape parameter :

Skewness is symmetric about the origin:

The limiting value of skewness is finite and depends on the sign of :

Kurtosis depends only on the shape parameter :

Kurtosis is symmetric about the origin and attains its minimum at 0:

The limiting value is larger than kurtosis of NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order is known for the zero location parameter:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

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:

Plot the distribution density:

Compute four standard moments for this group:

Compute the mean residual life function of a skew-normal random variate:

Plot the mean residual life function for several values of parameter , including the limiting case of normal variate, i.e., :

The finishing times for a 2010 Chicago marathon follow a SkewNormalDistribution:

Porbability density function:

Find the average time:

Find under what time half of the finishers crossed the finish line:

The finishing time distribution is skewed to the right:

Parameter influence on the CDF for each :

Skew-normal distribution is closed under translation and scaling by a positive factor:

Relationships to other distributions:

NormalDistribution is a special case of SkewNormalDistribution:

SkewNormalDistribution is a transformation of normal distribution:

Probability density function can be expressed in terms of distribution functions of NormalDistribution:

HalfNormalDistribution is the limiting case of SkewNormalDistribution:

And for :

SkewNormalDistribution is a transformation of BinormalDistribution:

The largest component of the standardized binormal distribution follows SkewNormalDistribution:

The maximum of two variates with the same NormalDistribution follows a SkewNormalDistribution: