Probability across an interval using Poisson

Question

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

Answer 1

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 $\frac{15-10}{15} = \frac{1}{3}.$