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Topic: Prob that event A occurs before B...
Replies: 9   Last Post: Jun 25, 2013 5:06 AM

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JohnF

Posts: 150
Registered: 5/27/08
Re: Prob that event A occurs before B...
Posted: Jun 24, 2013 5:19 AM
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Paul <pepstein5@gmail.com> wrote:
> JohnF wrote:
>> JohnF wrote:
>>> You're given that two independent events, A and B, will occur
>>> at some future times. You don't know when they'll occur, but
>>> you're given two pdf's for that, a(t),b(t),t>=0, with all
>>> the usual interpretation. What's the prob A occurs before B?

>>
>> Thanks, but never mind, I got it.
>> in case anyone's interested...
>> Let the cumulative prob be A(t) = int_0^t a(t)dt, and
>> similarly for B(t). Then G(t) = A(t)(1-B(t)) is the prob
>> that at or before t, A has occurred and B hasn't,
>> but that doesn't work as the "kernel" for anything,
>> which was my original mistake. Instead, a(t)(1-B(t))dt is the prob
>> that A occurs between t and t+dt, and that B hasn't occurred
>> yet. So just integrate that, P_ab = int_0^infty a(t)(1-B(t))dt.
>> And note how P_ba = 1-P_ab.
>>
>>... And not sure what Paul Epstein's talking about, ...

>
> I started by saying this: "Imagine that the number of times
> the event can occur is large but finite (N). Solve this case
> and translate to your continuous case"
>
> it seems a bit pointless to ask for advice, and then reply
> by saying that you don't know what others are talking about,
> without asking for clarification.


Fair enough. When you (ambiguously) say "the event", I assume
you mean that both events A and B can each occur N times, and that
my original pdf's a(t) and b(t) are now frequency distributions,
normalized to N rather than to 1.
But then, how does my question "what's the prob that
A occurs before B?" translate to this discrete scenario,
where A's and B's are each occurring all the time (except
for the trivial case where the distributions are disjoint,
with all A's occurring before any B's)?
So I don't see how you can even set up the problem,
much less solve it, i.e., "not sure what you're talking
about", in my original words.

> Surely, it's excellent advice to translate a continuous problem
> into a discrete problem and then solve the discrete problem.


Actually, I'd have liked to see how to do this (to xlate to
discrete problem), and tried to figure out what you were
talking about, but couldn't. Could you clarify, in light
of above? Thanks,
--
John Forkosh ( mailto: j@f.com where j=john and f=forkosh )



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