BetaDistribution

Generate a set of pseudorandom numbers that are beta distributed:

Compare the 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 varies with shape parameters:

When both parameters go to , the distribution becomes symmetric:

Kurtosis varies with shape parameters:

In the limit the kurtosis becomes the same as for NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Cloud duration approximately follows a beta distribution with parameters 0.3 and 0.4 for a particular location. Find the probability that cloud duration will be longer than half a day:

Simulate the fraction of the day that is cloudy over a period of one month:

Find the average cloudiness duration for a day:

Find the probability of having exactly 20 days in a month with cloud duration less than 10%:

Find the probability of at least 20 days in a month with cloud duration less than 10%:

Beta distribution can be used to model the proportion of the stocks that increase in value on a given day. Fit beta distribution into Dow Jones Industrial stocks:

Find daily change:

Filter out missing data and pad with zeros:

Calculate the daily ratio of companies with an increase in value:

Find fit, excluding days with zero companies having an increase in value:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the probability that at least 60% of Dow Jones Industrial stocks will increase in value:

Find the average percentage of Dow Jones Industrial stocks that will increase in value:

Simulate the percentage of Dow Jones Industrial stocks that will increase in value for 30 days:

Parameter influence on the CDF for each :

If a variate follows beta distribution, then follows the reflected distribution:

Relationships to other distributions:

BetaDistribution is equivalent to UniformDistribution:

BetaDistribution is a limiting case of NoncentralBetaDistribution:

BetaPrimeDistribution can be obtained as a transformation of the beta-distributed variable:

Beta distribution is a special case of PearsonDistribution of type 1:

Beta distribution can be obtained as a transformation of GammaDistribution:

Beta distribution can be obtained as a transformation of ChiSquareDistribution:

FRatioDistribution can be obtained from beta distribution:

Beta distribution is an order distribution of variables from UniformDistribution:

ExponentialDistribution is a limit of a scaled beta distribution:

KumaraswamyDistribution is a transformation of beta distribution:

KumaraswamyDistribution simplifies to a special case of beta distribution:

PERTDistribution is a transformation of beta distribution:

Univariate marginals of DirichletDistribution have beta distribution:

BetaBinomialDistribution is a mixture of BinomialDistribution and BetaDistribution:

BetaNegativeBinomialDistribution is a mixture of NegativeBinomialDistribution and BetaDistribution:

BetaDistribution is not defined when either or is not a positive real number:

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