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# SkewNormalDistribution

 SkewNormalDistribution represents a skew-normal distribution with shape parameter , location parameter , and scale parameter .
• The probability density for value in a skew-normal distribution is proportional to .
Probability density function:
Cumulative distribution function:
Mean and variance:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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 Scope   (7)
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:
Closed form for symbolic order is known for the zero location parameter:
Hazard function:
Quantile function:
 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:
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:
SkewNormalDistribution is a transformation of normal distribution:
Probability density function can be expressed in terms of distribution functions of NormalDistribution:
And for :
The largest component of the standardized binormal distribution follows SkewNormalDistribution:
The maximum of two variates with the same NormalDistribution follows a SkewNormalDistribution:
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