with G(x) denoting a function. That is, this is the beta cumulative distribution function, but the upper limit of integration is defined by another cumulative distribution function.
The case where G(x) denotes the normal cumulative distribution function results in the beta-normal distribution with the following cumulative distribution function:
with denoting the cumulative distribution function of the standard normal distribution.
This distribution can be extended with location and scale parameters by replacing the standard normal distribution with a normal distribution with location parameter, , and scale parameter, . The and are also the location and scale parameter of the beta-normal distribution.
where <x> is a number, parameter, or variable; <y> is a variable or a parameter (depending on what <x> is) where the computed beta-normal cdf value is stored;
<alpha> is a number, parameter, or variable that specifies the first shape parameter;
<beta> is a number, parameter, or variable that specifies the second shape parameter;
<loc> is a number, parameter, or variable that specifies the location parameter;
<scale> is a number, parameter, or variable that specifies the scale parameter;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
The <loc> and <scale> parameters are optional.
LET X2 = BNOCDF(X1,0.1,0.1)
PLOT BNOCDF(X,0.1,0.1) FOR X = -10 0.01 10
Eugene, Lee, and Famoye (2002). "Beta-Normal Distribution and Its Applications", Communications in Statistics-Theory and Methods, 31, pp. 497-512.
MULTIPLOT 3 3 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 3 LABEL CASE ASIS X1LABEL X Y1LABEL Probability Y1TIC MARK LABEL DECIMAL 2 X1LABEL DISPLACEMENT 14 Y1LABEL DISPLACEMENT 15 TITLE DISPLACEMENT 2 . LET A = DATA 0.1 1 5 LET B = DATA 0.1 1 5 LOOP FOR K = 1 1 3 LET ALPHA = A(K) LOOP FOR L = 1 1 3 LET BETA = B(L) TITLE ALPHA = ^ALPHA, BETA = ^BETA PLOT BNOCDF(X,ALPHA,BETA) FOR X = -10 0.01 10 END OF LOOP END OF LOOP . END OF MULTIPLOT JUSTIFICATION CENTER MOVE 50 97 CASE ASIS TEXT CDF's For Beta-Normal Distribution
Date created: 3/27/2006