Re: [Maxima-discuss] "distrib" package

[Robert D. <rob...@gm...>]

On 2014-09-03, Henry Baker <hb...@pi...> wrote:

> Thus, for country A, with population Ap, mean Am, variance Av, the
> abilities of the individuals follow a normal distribution.
>
> Clearly, each country will choose from the upper tail of the
> distribution -- its highest 1%, for example.
>
> What is the probability that a 1% athlete from A will beat a
> 1% athlete from B, given the population, mean and variance of
> each country ?

OK, I've derived an exact result for the distribution of ability(A)
minus ability(B) where ability(A) is a Gaussian tail, likewise
ability(B). I can explain the black art by which it was devised
but not at the moment.

But that's applicable only if all you know is that the atheletes
are the stars of some country or the other. If you have more
specific data, such as the recorded scores in the same event,
that should be the distribution of ability, not the population tail.
My guess is that individual scores are probably pretty much
Gaussian, so figuring P(ability(A) > ability(B)) becomes easy again.

best

Robert Dodier

PS.
Here's the expression for the distribution of u = ability(A) minus
ability(B) where A and B are from the tails of Gaussian distributions
with mean m1, m2, std dev s1, s2, and cut-off a1, a2, respectively.

(sqrt(2)*sqrt(%pi)*s1*s2*sqrt(s2^2+s1^2)
    *%e^(s1^2*u^2/(2*s2^4+2*s1^2*s2^2)-u^2/(2*s2^2)
                                      +m2*s1^2*u/(s2^4+s1^2*s2^2)
                                      +m1*u/(s2^2+s1^2)-m2*u/s2^2
                                      +m2^2*s1^2/(2*s2^4+2*s1^2*s2^2)
                                      +m1^2*s2^2/(2*s1^2*s2^2+2*s1^4)
                                      +m1*m2/(s2^2+s1^2)-m2^2/(2*s2^2))
    *erf(sqrt(s2^2+s1^2)*(sqrt(2)*s1^2*u
                         +(sqrt(2)*m1-sqrt(2)*max(a1,u+a2))*s2^2
                         +(sqrt(2)*m2-sqrt(2)*max(a1,u+a2))*s1^2)
          /(2*s1*s2^3+2*s1^3*s2))
 /(2*%e^(m1^2/(2*s1^2))*s2^2+2*s1^2*%e^(m1^2/(2*s1^2)))
 +sqrt(%pi)*s1*s2
       *%e^(s1^2*u^2/(2*s2^4+2*s1^2*s2^2)-u^2/(2*s2^2)
                                         +m2*s1^2*u/(s2^4+s1^2*s2^2)
                                         +m1*u/(s2^2+s1^2)-m2*u/s2^2
                                         +m2^2*s1^2/(2*s2^4+2*s1^2*s2^2)
                                         +m1^2*s2^2/(2*s1^2*s2^2+2*s1^4)
                                         +m1*m2/(s2^2+s1^2)-m2^2/(2*s2^2)
                                         -m1^2/(2*s1^2))
  /(sqrt(2)*sqrt(s2^2+s1^2)))
 /(2*%pi*Z(a1,m1,s1)*Z(a2,m2,s2)*s1*s2)$