Re: P(m,n)=C(m,n)*p^m*q^(n-m) generalization

On Feb 9, 1:23 pm, quasi <qu...@xxxxxxxx> wrote:
On Mon, 09 Feb 2009 11:24:45 EST, maxim <max...@xxxxxxx> wrote:
how to generalize formula P(m,n)=C(m,n)*p^m*q^(n-m) if random variables

Which random variables?

are correlated (assume ro=const)?

Your problem is not stated in a precise way.

However, I'll make a few guesses as to what might have been intended.

Presumably, it's a binomial experiment -- coin flipping, for example,
but where the probability of heads depends on the outcome of the
previous flip, hence a correlation.

Thus, let p1 = probability of heads, given that the previous was a
flip was a head. and let p2 = probability of heads given that the
previous flip was a tail. Let's assume p1, p2 are constant. I think
the first thing to do is solve for the conditional probabilities p1,
p2, in terms of rho. However there are 2 unknowns, p1 and p2, and only
one restriction, namely, the given value of rho. To reduce the number
of variables, you need a simplifying assumption, something that
connects p1 and p2.

Another issue is the starting probability. The simplest way to deal
with the first flip is to is to assume the previous flip is given
information.

Then, for fixed p1,p2 there will be 2 functions P(m,n)

   P(m,n|H) = probability of m heads in n flips, given that
   the zero'th flip is a head.

   P(m,n|T) = probability of m heads in n flips, given that
   the zero'th flip is a tail.

Note that the zero'th flip is not part of the count -- it's just
initial information, so that the probabilities for the first flip are
determined.

I'm not sure if it will be possible, without some further assumptions,
to get a closed form for P(m,n|H) and P(m,n,T).

However it seems clear that you could model it as a Markov chain with
states represented by triples (h,k,L), where k is the number of flips
so far, h is the number of heads in those flips, and L is the result
of the last flip (either H or T). The initial state is either (0,0,T)
or (0,0,H), and it would have to be specified as to which one. The
terminating states are all states of the form (h,n,L) -- in other
words, all states with k = n. Then P(m,n|H) is the probability of
terminating in state (m,n,T) or (m,n,H) given initial state (0,0,H).
Similarly, P(m,n|T) is the probability of terminating in state (m,n,T)
or (m,n,H) given initial state (0,0,T).

For given p1,p2,m,n, you should be able to calculate P(m,n|H) and
P(m,n|T) exactly using the transition probabilities of the Markov
chain.

Alternatively, you can always do a simulation to get approximate
values.

If I've misinterpreted your question -- sorry. But then again, you
didn't give a very clear specification of the problem.

quasi

the troll of the sci, physics and math threads, vliad, almost, karlos

the following is my opinion:
the eTroll is William O'Leary, b. 1964 of Austin , TX.
he haunts alt.rainbow.gathering
as Eliiyahu Simchah, Tha Preacha Bill, and also
.
Karlos, Almost, Vliad, and his latest names MARCOS and RITA
He's more than an eTroll. trolling is just a pastime.

his real interest is full-on cyberstalking.
when he can find a vulnerable enough victim, he will confront them in
real
life, but only if they are elderly, disabled, or a woman.

he was fired from his job in december because of this.
that's pretty tough, because this is the first time he's had a job
in a couple of years.

since 1986 he's been homeless, in a shelter, couch-surfing,
in several states.

or most recently Urban Camping in austin, in a tent, pooping
in buckets on a hippie "farm"

he illegaly represents himself as a paralegal.

here's a link with pdf files about his recent firing due to
cyberstalking on the job.

there;'s also a horrifying MP4 video file,
which shows him harrassing and apparently physically threatening a
disabled woman and her BF in her home.

the camera operator is on an oxygen bottle. her BF is in the image.

he was ultimately removed by the cops.

http://sites.google.com/site/williamolearysite
.

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