ALPCDF

ALPCDF (LET)

Library Function

Compute the alpha cumulative distribution function with shape parameter .

The alpha distribution has the following cumulative distribution function:

with denoting the shape parameter.

This distribution can be generalized with location and scale parameters using the relation

If Y has a normal distribution with location and scale parameters and truncated to the left of 0, then X = 1/Y has an alpha distribution with shape parameter = / and scale parameter .

This distribution has application in reliability.

LET <y> = ALPCDF(<x>,<alpha>,<loc>,<scale>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable containing positive values;
            <y> is a variable or a parameter (depending on what <x> is) where the computed alpha cdf value is stored;
            <alpha> is a positive number, parameter, or variable that specifies the shape parameter;
            <loc> is a number, parameter, or variable that specifies the location parameter;
            <scale> is a positive number, parameter, or variable that specifies the scale parameter;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If <loc> and <scale> are omitted, they default to 0 and 1, respectively.

LET A = ALPCDF(0.3,2.5)
LET A = ALPCDF(X1,2.5,0,10)
PLOT ALPCDF(X,2.5,0,3) FOR X = 0.1 0.1 10

The 11/2007 version changed the syntax for this function from
LET A = ALPCDF(X,ALPHA,BETA,LOC,SCALE)

to

LET A = ALPCDF(X,ALPHA,LOC,SCALE)

This was done since BETA is in fact a scale parameter.

None

None

ALPPDF	= Compute the alpha probability density function.
ALPPPF	= Compute the alpha percent point function.
ALPHAZ	= Compute the alpha hazard function.
ALPCHAZ	= Compute the alpha cumulative hazard function.
PEXPDF	= Compute the exponential power probability density function.
WEIPDF	= Compute the Weibull probability density function.
LGNPDF	= Compute the log-normal probability density function.
NORPDF	= Compute the normal probability density function.
NORCDF	= Compute the normal cumulative distribution function.

Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions--Volume 1", Second Edition, John Wiley and Sons, p. 173.

Salvia (1985), "Reliability applications of the Alpha Distribution", IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 251-252.

Reliability, accelerated life testing

1998/4
2007/11: Corrected the second shape parameter to be the scale parameter

LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
.
LET ALPHA  = 0.5
TITLE ALPHA = ^alpha
PLOT ALPCDF(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 1
TITLE ALPHA = ^alpha
PLOT ALPCDF(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 2
TITLE ALPHA = ^alpha
PLOT ALPCDF(X,ALPHA) FOR X = 0.01  0.01  5
.
LET ALPHA  = 5
TITLE ALPHA = ^alpha
PLOT ALPCDF(X,ALPHA) FOR X = 0.01  0.01  5
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Alpha Cumulative Distribution Functions