betaCdf¶

Evaluates the beta probability distribution function.

Synopsis¶

betaCdf (x, pin, qin)

Required Arguments¶

float x (Input): Argument for which the beta probability distribution function is to be evaluated.
float pin (Input): First beta distribution parameter. Argument pin must be positive.
float qin (Input): Second beta distribution parameter. Argument qin must be positive.

Return Value¶

The probability that a beta random variable takes on a value less than or equal to x.

Description¶

Function betaCdf evaluates the distribution function of a beta random variable with parameters pin and qin. This function is sometimes called the incomplete beta ratio and with p = pin and q = qin, is denoted by \(I_x(p,q)\). It is given by

\[I_x(p,q) = \frac{\mathit{\Gamma}(p) \mathit{\Gamma}(q)}{\mathit{\Gamma}(p+q)} \int_0^x t^{p-1} (1-t)^{q-1} dt\]

where \(\Gamma(\cdot)\) is the gamma function. The value of the distribution function by \(I_x(p,q)\) is the probability that the random variable takes a value less than or equal to x.

The integral in the expression above is called the incomplete beta function and is denoted by \(\beta_x(p,q)\). The constant in the expression is the reciprocal of the beta function (the incomplete function evaluated at one) and is denoted by \(\beta(p,q)\).

Function betaCdf uses the method of Bosten and Battiste (1974).

Example¶

Suppose X is a beta random variable with parameters 12 and 12. (X has a symmetric distribution.) This example finds the probability that X is less than 0.6 and the probability that X is between 0.5 and 0.6. (Since X is a symmetric beta random variable, the probability that it is less than 0.5 is 0.5.)

from __future__ import print_function
from numpy import *
from pyimsl.math.betaCdf import betaCdf

pin = 12.0
qin = 12.0
x = 0.6
p = betaCdf(x, pin, qin)
print("The probability that X is less than 0.6 is %6.4f" % (p))

x = 0.5
p = p - betaCdf(x, pin, qin)
print("The probability that X is less than between 0.5 and 0.6 is %6.4f"
      % (p))

Output¶

The probability that X is less than 0.6 is 0.8364
The probability that X is less than between 0.5 and 0.6 is 0.3364