Poisson processContd...Example 4: The reporting of customers at a counter in a railway station is a Poisson random process with mean rate of 3 customers in a minute. Find the probability that the interval of time between two consecutive customers is (i) more than a minute, (ii) between 1 and 2 minutes, (ii) at the most 3 minutes.

The Poisson process is often used to model the number of times an event occurs in an interval of time or space. In this case, we are given that the mean rate of customers arriving at a counter is 3 per minute. This is our λ (lambda) value.

(i) To find the probability that the interval of time between two consecutive customers is more than a minute, we need to find the probability that no customers arrive in a one-minute interval. In a Poisson process, the probability of zero events in an interval is given by the formula P(X=0) = e^(-λ). Here, λ = 3, so:

P(X=0) = e^(-3) ≈ 0.0498

So, the probability that the interval of time between two consecutive customers is more than a minute is approximately 0.0498, or 4.98%.

(ii) To find the probability that the interval of time between two consecutive customers is between 1 and 2 minutes, we need to find the probability that exactly one customer arrives in a one-minute interval, and then subtract this from the probability that at most one customer arrives in a two-minute interval. The probability of exactly one event in an interval is given by the formula P(X=1) = λe^(-λ), and the probability of at most one event in an interval is given by the formula P(X≤1) = e^(-λ)(1+λ). Here, λ = 3 for a one-minute interval and λ = 6 for a two-minute interval, so:

P(X=1) = 3e^(-3) ≈ 0.1494 P(X≤1) = e^(-6)(1+6) ≈ 0.0025

So, the probability that the interval of time between two consecutive customers is between 1 and 2 minutes is approximately 0.0025 - 0.1494 = -0.1469. However, probabilities cannot be negative, so this result indicates that the probability is essentially zero.

(iii) To find the probability that the interval of time between two consecutive customers is at most 3 minutes, we need to find the probability that at least one customer arrives in a three-minute interval. This is given by 1 - P(X=0), where λ = 9 for a three-minute interval. So:

P(X=0) = e^(-9) ≈ 0.0001 1 - P(X=0) ≈ 1 - 0.0001 = 0.9999

So, the probability that the interval of time between two consecutive customers is at most 3 minutes is approximately 0.9999, or 99.99%.