Solved – How to find a confidence interval for the total number of events

confidence intervalprobability

I have detector which will detect an event with some probability p. If the detector says that an event occured, then that is always the case, so there are not false-positives. After I run it for some time, I get k events detected. I would like to calculate what the total number of events that occured was, detected or otherwise, with some confidence, say 95%.

So for example, let's say I get 13 events detected. I would like to be able to calculate that there were between 13 and 19 events with 95% confidence based on p.

Here's what I've tried so far:

The probability of detecting k events if there were n total is:

binomial(n, k) * p^k * (1 - p)^(n - k)

The sum of that over n from k to infinity is:

1/p

Which means, that the probability of there being n events total is:

f(n) = binomial(n, k) * p^(k + 1) * (1 - p)^(n - k)

So if I want to be 95% sure I should find the first partial sum f(k) + f(k+1) + f(k+2) ... + f(k+m) which is at least 0.95 and the answer is [k, k+m]. Is this the correct approach? Also is there a closed formula for the answer?

Best Answer

I would choose to use the negative binomial distribution, which returns the probability that there will be X failures before the k_th success, when the constant probability of a success is p.

Using an example

k=17 # number of successes
p=.6 # constant probability of success

the mean and sd for the failures are given by

mean.X <- k*(1-p)/p
sd.X <- sqrt(k*(1-p)/p^2)

The distribution of the failures X, will have approximately that shape

plot(dnbinom(0:(mean.X + 3 * sd.X),k,p),type='l')

So, the number of failures will be (with 95% confidence) approximately between

qnbinom(.025,k,p)
[1] 4

and

qnbinom(.975,k,p)
[1] 21

So you inerval would be [k+qnbinom(.025,k,p),k+qnbinom(.975,k,p)] (using the example's numbers [21,38] )

Related Question