Re: How to calculate this probability ditribution ?

Dear Koopman,

Thabnks for you answer,
I think a NegativeBinomial distribution may be right with your event
definition of "s Or t",
but I am wondering that in the 2x2 table,
t nt
s a b
ns c d
if a,b,c,d are all clearly defined, (a+b+c+d=1), among them d is the
probability of no-signal no-triger. An event is not defined as "s Or
t", but defined by all the occurence of s-t,s-nt,ns-t and ns-nt, then
how about the probability distribution of s-nt before Nth of t (by s-t
or ns-t) ?
In this case, the distribution of M = nt's number before Nth t, will
be NegativeBinomial, but M includes s-nt or ns-nt, not only s-nt. So
let n= the number of s-nt, it will
follow a Binomial distribution (M,n,b/(b+d)), but here M is not a
constant, but a random variale with a NegativeBinomial distribution,
the final distribution of n will be complex, is it right ?

Or can I still take d as irrelevant in this case? If so, I can still
use the your calculation with the event definition of "s Or t" .

Hope you advice, thanks a lot.

Wang

PS: Time is not involved in this discussion.

Ray Koopman 写道：

Ray Koopman wrote:
Xufei.Wang@xxxxxxxxx wrote:
Dears


I am working on a problem of probability disribution as following:

For a real pulse fed into a discrimiantor with preset threshold for
counting the number, the probability of it having enough height to
triger the discimiator is q.
Noises are also inevitably go into the dicrinmimator, and for a
trigered count in the discriminator, the probability of it being
trigered by a real pulse is p, and the probability of being trigered by

a noise signal is (1-p).

The question is:
Given N counts (possibly including real pulses and noises ) are
trigered in the discriminator, how to calculate the distribution of the

missed number of real pulses which have the lower height not enough
to triger the discriminator ?


Thank you.

This appears to be an attempt to clarify your May 25 sci.math post
"How to calculate this probability?". In the implied 2x2 table:

t nt
s a b
ns c d

where a,b,c,d are probabilities (a + b + c + d = 1),
is d well defined? Does it make sense to talk about the probability
of a no-signal, no-trigger event? Is an "event" defined by the
occurrence of a signal and/or a trigger? Or is time involved somehow?

If an event is defined as "s OR t" then d is irrelevant.

We have q = a/(a+b) <=> b = (1/q - 1)a

and p = a/(a+c) <=> c = (1/p - 1)a.

a + c a + (1/p - 1)a 1/p
Let r = --------- = --------------------------- = -------------
a + c + b a + (1/p - 1)a + (1/q - 1)a 1/p + 1/q - 1

and let M = the number of nt's before the Nth t.

Then M ~ NegativeBinomial[N,r]:

P(M = m) = r^N (1-r)^m (N-1+m)!/((N-1)!m!), m = 0,1,2,... .

.

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