Re: How to evaluate the propability distribution of two dices?

Dear Andrey,

Hmm, so the dice have different probability distributions, and we're
not sure
that every possible outcome appears in the sample. That makes it
difficult. I
don't know the answer, either; perhaps somebody else does?

Another possibility. We have again the same grid. (I like to visualise
things.)

1 2 3 4 5 6 (d2)
(d1) +------------
1 | 2 3 4 5 6 7
2 | 3 4 5 .
3 | 4 5 6 .
4 | 5 . .
5 | 6 . .
6 | 7 . . . .12

Let f-hat be the predicted frequency of the various outcomes. We then
have
the following set of equations:

f-hat(sum= 2) = N * ( p(d1=1)*p(d2=1) )
f-hat(sum= 3) = N * ( p(d1=1)*p(d2=1) + p(d1=2)*p(d2=1) )
f-hat(sum= 4) = N * ( p(d1=1)*p(d2=3) + p(d1=2)*p(d2=2) +
p(d1=3)*p(d2=1) )
....
f-hat(sum=12) = N * ( p(d1=6)*p(d2=6) ) .

In addition, we have the two constraints

1 = p(d1=1) + . . . + p(d1=6) and
1 = p(d2=1) + . . . + p(d2=6) .

That's 13 equations with 12 unknowns. Let f be the observed
frequencies. We
want to choose all the p-values (probability densities) in such a way
as to
minimise the sum of squares, i.e. the sum of (f-hat - f)^2.

So, that makes it a problem of optimisation under constraints. You're
right,
though: we get difficult non-linear equations. Hopefully you can feed
this to
a computer programme to solve.

The sum of squares can then be used to quantify --- use the F
statistic to see
whether we have significant good fit, and the R^2 statistic to see how
much
variance we explain with this model. (If you don't know how to compute
F and
R^2, search for "F statistic" (or "R statistic") + "by hand". That
should find
you a good intro. If that doesn't help, shoot me a e-mail in private:
sbbrouwer#gmail,com.)

By the way: might I ask what you need this for? It seems like an
interesting
course, if it is a course.

Cheers,

Sietse
Sietse Brouwer

.

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