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Working through some Poisson probability questions and am wondering whether the knowledge of how many events have occurred changes a calculation.

The question is thus:

The number of prank calls received by an bartender is modeled using a Poisson process with rate 3 per hour. If we consider a time interval and we know the exact number of calls received in that time interval, then the calls are distributed independently and uniformly within that time interval. We know that exactly one call was received from 10pm to 10.15pm: what is the probability that this call was received after 10.10pm?

Normally my approach would have been the following: $ λ = \frac {60}{20} $ across whole interval

Poisson PMF: $\frac {(e^{-λ})*λ^x)}{x!}$

$ λ = \frac {10}{20} $ for the fraction of the interval we are interested in

PMF = $1-\frac {(e^{-.5})*.5^1)}{1}$

PMF = $1-.303$

PMF = $.697$

However the questions states that 'we know that one call was received from 10pm to 10.15pm'. Does this change the λ from 3 to 4? Or should I be looking at an exponential distribution instead - eg. $ fx(x) = e^{(−λx)}$?.

Thanks in advance

Regards,

Greg

Frank Anderson
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1 Answers1

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You're told that the conditional distribution of the event time, given occurrence in a fixed interval, is uniform:

If we consider a time interval and we know the exact number of calls received in that time interval, then the calls are distributed independently and uniformly within that time interval.

That means, if exactly one call was received between 10:00 and 10:15 PM, then the probability it occurred after 10:10 PM is simply $\frac{15-10}{15} = \frac{1}{3}.$

heropup
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  • I understand your logic - but theoretically I should be able to prove that mathematically using a distribution - and I can't seem to hit the 1/3. – Frank Anderson Feb 24 '21 at 01:49
  • In fact - if you were to use an exponential CDF - you would get a much different answer - lambda = 1/15, t = 10, 1- e^-10/15 = .49 – Frank Anderson Feb 24 '21 at 02:12
  • @GregWiseman Using an exponential distribution is incorrect because you are told exactly one event happens in a given time interval. Specifically, you are told the event happens in $15$ minutes with probability $1$, but your logic would mean the probability is $1 - e^{-15/15} < 1$. – heropup Feb 24 '21 at 02:21