Question

Consider a poisson arrival process with rate lambda per hour. To simplify notation, we let a=P(0,1), b=P(1,1) and c=P(2,1), where P(k,1) is the probability of exactly k arrivals over an hour-long time interval.

What is the probability that we will have almost one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 ad 12:00? Your answer should be an algebraic function of alb and c.

Answer 1

To find the probability that we will have almost one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00, we can use the properties of a Poisson process.

First, let's consider the probability of having almost one arrival between 10:00 and 11:00. This means there can be either zero or one arrival during this time interval. Therefore, the probability of having almost one arrival between 10:00 and 11:00 is given by:

P(almost one arrival between 10:00 and 11:00) = P(0,1) + P(1,1)
= a + b

Next, let's consider the probability of having exactly two arrivals between 10:00 and 12:00. This means there can be two arrivals during the entire 2-hour time interval. Therefore, the probability of having exactly two arrivals between 10:00 and 12:00 is given by:

P(exactly two arrivals between 10:00 and 12:00) = P(2,2)
= c^2

To find the joint probability of both events occurring, we simply multiply the individual probabilities:

P(almost one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00) = (a + b) * c^2

Therefore, the probability we are looking for is an algebraic function of a, b, and c given by:

P(almost one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00) = (a + b) * c^2