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Beta distribution

Beta Distribution

Probability Density FunctionThe general formula for the probability density function of the beta distribution isf(x) = (x-a)^(p-1)*(b-x)^(q-1)/(B(p,q)*(b-a)^(p+q-1)) for a <= x <= b, p, q > 0
where p and q are the shape parameters, a and b are the lower and upper bounds, respectively, of the distribution, and B(p,q) is the beta function. The beta function has the formula
B(alpha,beta) = INTEGRAL[0 to 1][t^(alpha-1)*(1-t)^(beta-1)dt]
The case where a = 0 and b = 1 is called the standard beta distribution. The equation for the standard beta distribution is
f(x) = x^(p-1)*(1-x)^(q-1)/B(p,q) for 0 <= x <= 1, p, q > 0
Typically we define the general form of a distribution in terms of location and scale parameters. The beta is different in that we define the general distribution in terms of the lower and upper bounds. However, the location and scale parameters can be defined in terms of the lower and upper limits as follows:
    location = a
    scale = b - a
Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.The following is the plot of the beta probability density function for four different values of the shape parameters.
plot of the Beta probability density function for 4 different values of the shape parameters
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the beta distribution is also called the incomplete beta function ratio (commonly denoted by Ix) and is defined as
    F(x) = I(x)(p,q) = (1/B(p,q))*INTEGRAL[0 to x][t^(p-1)*(1-t)^(q-1) dt
where B is the beta function defined above.The following is the plot of the beta cumulative distribution function with the same values of the shape parameters as the pdf plots above.
plot of the Beta cumulative distribution function with the same values of the shape parameters as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the beta distribution does not exist in a simple closed form. It is computed numerically.The following is the plot of the beta percent point function with the same values of the shape parameters as the pdf plots above.
plot of the beta percent point function with the same values of the shape parameters as the pdf plots above
Other Probability FunctionsSince the beta distribution is not typically used for reliability applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common StatisticsThe formulas below are for the case where the lower limit is zero and the upper limit is one.
Meanp/(p+q)
Mode(p-1)/(p+q-2) p, q > 1
Range0 to 1
Standard DeviationSQRT[(p*q)/((p+q)^2*(p+q+1))]
Coefficient of VariationSQRT(q/(p*(p+q+1)))
SkewnessSQRT[2*(q-p)*(SQRT(p+q+1)]/[(p+q+2)*SQRT(p*q)]
Parameter EstimationFirst consider the case where a and b are assumed to be known. For this case, the method of moments estimates are
    p = xbar*[(xbar*(1-xbar)/s^2) - 1]p = xbar*[(xbar*(1-xbar)/s^2) - 1]
where xbar is the sample mean and s2 is the sample variance. If a and b are not 0 and 1, respectively, then replace xbar with (xbar-a)/(b-a) and s2 with s^2/((b-a)^2) in the above equations.For the case when a and b are known, the maximum likelihood estimates can be obtained by solving the following set of equations
    psi(phat)-psi(phat+qhat)=(1/n)*SUM[i=1 to n][LOG[(Y(i)-b)/(b-a)]]psi(phat)-psi(phat+qhat)=(1/n)*SUM[i=1 to n][LOG[(Y(i)-b)/(b-a)]]
The maximum likelihood equations for the case when a and b are not known are given in pages 221-235 of Volume II of Johnson, Kotz, and Balakrishan.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the beta distribution.

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