Re: 4 parameter beta equivalence


Samik R. wrote:
On 10/6/2006 10:03 AM, Lou Thraki wrote:
Samik R. wrote:
Hello,

I have a seemingly simple question - though I wasn't able to solve
myself. Using Beta(min, max, alpha, beta) as the notation for the
4-parameter beta distribution, the following two same?
1. Beta(min, max, alpha, beta)
2. Beta(0,1,alpha,beta) * (max-min) + min

Most will probably realize that the second one is the formula for PERT
distribution used in project management.

My algebraic manipulations seemed to lead to the following:
PDF of (1):
(x - min)^(alpha-1) * (max - x)^(beta-1)
----------------------------------------
B(alpha,beta) * (max - min)^(alpha+beta-2)

PDF of (2):
x^(alpha-1) * (1-x)^(beta-1)
----------------------------- * (max - min) + min
B(alpha,beta)

If you generate x in [0,1] following Beta(0,1,alpha,beta),
and you perform the transformation y = (max-min)*x + min,
then the variable y will be distributed in [min,max]
following PDF (1).


Does that mean they are equivalent?

(2) is not a PDF

Rephrasing, if I generate a set of 100 uniform random numbers between
(0,1), and pass them through the inverse CDF of (1) and (2), will I get
the same values?

If you pass uniform random numbers between (0,1) through the
inverse CDF of Beta(0,1,alpha,beta), and then perform the
transformation I suggested, you will get the same result as when
you pass the uniform random numbers between (0,1) through
the inverse CDF of (1).

.

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