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Problem 1.3. As a complication of the Rutherford–Chadwick–Ellis experiment, suppose that the number of particles that decay in each interval is registered by a counter. We assume that events occur according to a Poisson process with intensity λ > 0, but whenever the counter registers an event, it is inoperative for the next b ≥ 0 units of time (and does not register any new events in that time interval). Let R(t) denote the number of events that are registered by time t ≥ 0.

(i) What is the probability that the first k events are all registered, for each k = 1, 2, . . .?

(ii) For t ≥ (n − 1)b, find P{R(t) ≥ n}.

Many thanks! first part of my solution

second part of my solution

appleyuchi
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1 Answers1

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After an event is registered, the probability that no event occurs during the latency period is $\mathrm e^{-\lambda b}$. The probability for this to happen $k$ times in a row is $\mathrm e^{-k\lambda b}$.

For $t\ge(n-1)b$, there is enough time for $n-1$ latency periods, and they take up a time $(n-1)b$, leaving $t-(n-1)b$. The probability for at least $n$ events to occur in this time is

$$ \mathrm e^{-\mu}\sum_{k=n}^\infty\frac{\mu^k}{k!}=1-\mathrm e^{-\mu}\sum_{k=0}^{n-1}\frac{\mu^k}{k!} $$

with $\mu=\lambda(t-(n-1)b)$.

joriki
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  • thanks a lot,but I want to know that why can not I use poisson distribution to calculate (a)? I mean...I regard "b unit time" as the time in poisson distribution,then ,the first event happen is λe^(-λ),afterwards,I consider the followding k-1 events – appleyuchi Sep 21 '18 at 09:50
  • @appleyuchi: You might be able to do that, I haven't tried it -- but it seems like a hell of a lot of unnecessary work when such a simple solution is available, no? – joriki Sep 21 '18 at 09:58
  • My answer for (a)is:[Σe^(-λj)·λe^(-λ)] · [Σe^(-λm)·λe^(-λ)]^(k-1) while j∈[0,∞],and m∈[b,∞],could you point it out where it is wrong please?Many Thanks. – appleyuchi Sep 21 '18 at 10:02
  • I mean,I regard the word of "unit" in "b unit time" as t=1 in poisson probability formula P{N(t)=k}=>P{N(1)=k}@joriki – appleyuchi Sep 21 '18 at 10:06
  • @appleyuchi: Please see how to ask a good question. If you already had an answer, you should have posted it along with the question. The site isn't meant to work like this, with long comments containing a major part of the content of the question and the answer. – joriki Sep 21 '18 at 10:31
  • I am not sure whether my answer is riht,so I did not post – appleyuchi Sep 21 '18 at 10:32
  • Also, please see this tutorial and reference on how to typeset math on this site. I find the stuff you wrote above too difficult to read. I'll try to answer your question if you add it to the main question in more readable formatting. – joriki Sep 21 '18 at 10:32
  • I'' try it now,thanks – appleyuchi Sep 21 '18 at 10:33
  • I have no rights to display image on this site ,could you have a look at my two links of solution please?It seems that I cannot master to use code to display formula in short time,thanks. – appleyuchi Sep 21 '18 at 11:00
  • @appleyuchi: I'm not in a hurry, take your time. – joriki Sep 21 '18 at 11:03
  • I have look for many sites ,it seems that no "convert formula to Mathjax code" websites on the internet,may be some websites are forbiddened to enter by my government,if I learn the whole grammer,it will takes several weeks,but I need to learn other knowledge. Many thanks again. – appleyuchi Sep 21 '18 at 11:19