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
. It is given by
where Γ (⋅) is the gamma function. This function is sometimes called the
incomplete beta ratio and, with p = pin
and q = qin
, is denoted
by \(I_x(p,q)\).
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 1) 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.stat.betaCdf import betaCdf
pin = 12.0
qin = 12.0
x = 0.6
pr1 = betaCdf(x, pin, qin)
print("The probability that X is less than 0.6 is %6.4f" % pr1)
x = 0.5
pr2 = pr1 - betaCdf(x, pin, qin)
print("The probability that X is between 0.5 and 0.6 is %6.4f" % pr2)
Output¶
The probability that X is less than 0.6 is 0.8364
The probability that X is between 0.5 and 0.6 is 0.3364