random.py gives wrong results (+ a solution)

[Tim Peters]

[Ivan Frohne]
> I'm convinced that Janne Sinkkonen is right: The beta distribution
> generator in module random.py does not return Beta-distributed random
> numbers.  Janne's suggested fix should work just fine.

Which is (for reference):

def betavariate(alpha, beta):
    y = gammavariate(alpha,1)
    if y==0: return 0.0
    else: return  y/(y+gammavariate(beta,1))

> Here's my guess on how and why this bug bit -- it won't be of interest
> to  most but this subject is so obscure sometimes that there needs to
> be a detailed analysis.
> ...

Which I'll skip, since it's immortalized already in c.l.py and the bug
report:

https://sourceforge.net/bugs/?func=detailbug&bug_id=130030&group_id=5470


> ...
> The corrected equation for a beta random variable, using Watkins'
> definition of the gamma density, is
>
>     brv(A, B) = grv(A, 1) / ( grv(A, 1) + grv(1/B, 1) ),
>
> which translates to
>
>     brv(A, B) = grv(A, 1) / (grv(A, 1) + grv(B, 1)
>
> using the more common gamma density definition (the one used in random.
> py).

Now that seems plain wrong, although I assume it's just typographical
confusions:

1. You don't really mean to generate two independent instances of grv(A, 1),
right?

2. The parens around the summand denominator have magically disappeared,
changing something of the form X/(X+Y) to (X/X)+Y.

> Many standard statistical references give this equation -- two are
> "Non-Uniform random Variate Generation," by Luc Devroye, Springer-Verlag,
> 1986, p. 432, and "Monte Carlo Concepts, Algorithms and Applications,"
> by George S. Fishman, Springer, 1996, p. 200.

So does Knuth, Vol 3 Ed 3 (in the X/(X+Y) form), although he's not careful
(as was Janne) to avoid division by 0.  So I'll check in Janne's code as-is.

Thank you for the analysis!  It was an enlightening help.