Question

When events occur in a time period (0, a) with regard to a Poisson process, it is well known that, conditioned on the total number of events k, the joint distribution of the times at with the events occur follows a uniform distribution in (0, a) k. That is, if Xi represents the arrival time of one of them, then Xi ∼ U(0, a) and it is independent of the other X’s.

1. If ten patients visit the Emergency Room (ER) of a hospital between 9 and 10, what is the probability that fifth of them reaches the ER before 9:30?

2. If ten patients visit the ER between 9 and 10, what is the probability that the first 3 of them arrive before 9:20, 4 of them between 9:20 and 9:40, and the last 3 of them after 9:40?

3. If two patients visit the ER of a hospital between 9 and 10, what is the conditional distribution of the time at which the second patient arrived at the ER given that the first patient arrived at time t1?

Answer 1

The arrival times   folows Uniform distribution with and are independent of each other.

1) If 10 patients visit the Emergency Room (ER) of a hospital between 9 and 10 then their arrival times is ditributed as . The probability that one patient arrive before 9:30 is

.

The number of patients who arrive before 9:30 out of 10 has binomial distribution with PMF

. Here .

The probability that fifth of them (10/5=2) reaches the ER before 9:30 is

2) Here we have to consider the trinomial distribution since we are considering 3 probabilities.

i) Arrival before 9:20 with probability .

ii) Arrival between 9:20 and 9:40 with probability .

iii) Arrival after 9:40 with probability ​.

The probability that the first 3 of them arrive before 9:20, 4 of them between 9:20 and 9:40, and the last 3 of them after 9:40 is

Note that this is an extension of binomial distribution used in part(1). in which we had 2 events. Here 3 events.

3) As given  if two patients visit the ER of a hospital between 9 and 10, the arrival times   folows Uniform distribution with and are independent of each other. Hence the arrival of one patient does not affect that of the second patient. So the conditional distribution is the same

as .