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.
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
To find the probability of having almost one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00, we need to consider the different possibilities.
First, let's break down the time interval from 10:00 to 12:00 into two smaller intervals:
1) From 10:00 to 11:00 (1-hour interval)
2) From 11:00 to 12:00 (1-hour interval)
Now, let's consider the probabilities for each scenario:
1) From 10:00 to 11:00:
To have almost one arrival (at least one arrival) during this time interval, we need to consider three possibilities:
- One arrival (P(1,1))
- Two arrivals (P(2,1))
- More than two arrivals (P(3,1) + P(4,1) + ...)
So, the probability of having almost one arrival between 10:00 and 11:00 can be expressed as:
P(X >= 1 between 10:00 and 11:00) = P(1,1) + P(2,1) + P(3,1) + ... = b + c + (P(3,1) + P(4,1) + ...)
2) From 11:00 to 12:00:
To have exactly two arrivals during this 1-hour interval, we can consider the following possibility:
- One arrival between 11:00 and 11:30 (P(1,0.5))
- One arrival between 11:30 and 12:00 (P(1,0.5))
So, the probability of having exactly two arrivals between 10:00 and 12:00 can be expressed as:
P(two arrivals between 10:00 and 12:00) = P(1,0.5) * P(1,0.5)
Finally, to find the overall probability, we need to multiply the probabilities of the two scenarios:
P(almost one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00) = (b + c + (P(3,1) + P(4,1) + ...)) * (P(1,0.5) * P(1,0.5))
Note: The expression can be simplified using the properties of the Poisson distribution, but the resulting algebraic function will still depend on the values of a, b, and c.