represents a continuous beta distribution with shape parameters α and β.


Background & Context


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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:


Scope  (8)

Generate a sample of pseudorandom numbers from a beta distribution:

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:





Hazard function:

Quantile function:

Consistent use of Quantity in parameters expands them into their numeric values:

Find the mean:

Applications  (3)

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 to the Dow Jones Industrial stocks data:

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 no 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:

Discrete-time Markov chain , where is the sequence of independent and identically distributed (iid) standard uniform random variables, and is the sequence of iid Bernoulli random variables with success probability of converges to stationary distribution BetaDistribution[p,1-p] for any initial condition such that :

Sample a realization of the Markov chain and discard the burn-in portion of the path:

Samples from the Markov chain are not independent and exhibit internal structure:

Compare the histogram of path values to the PDF of the Markov chain's stationary distribution:

Use path values to approximate an expectation:

Compare with the quadrature value:

Properties & Relations  (21)

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

Relationships to other distributions:

BetaDistribution[1,1] is equivalent to UniformDistribution[{0,1}]:

BetaDistribution is a transformation of UniformDistribution:

UniformDistribution is a transformation of BetaDistribution:

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:

ExponentialDistribution is a transformation of 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:

WignerSemicircleDistribution is a transformation of special 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:

Possible Issues  (2)

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

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

Introduced in 2007
Updated in 2016