The consecutive interarrival times of a certain arrival process are i.i.d. random variables that are equally likely to be 5, 10, or 15 minutes. Find the expected value of the length of the interarrival time seen by an observer who arrives at some particular time, unrelated to the history of the process.
11.66666666
To find the expected value of the length of the interarrival time, we need to consider the probabilities of each possible interarrival time and then calculate their average.
In this scenario, the consecutive interarrival times are equally likely to be 5, 10, or 15 minutes. Let's denote these three possibilities as X1, X2, and X3, respectively.
To find the expected value, we can use the formula:
E(X) = Σ(xi * P(xi))
Where E(X) is the expected value, xi is the value of the random variable, and P(xi) is the probability of that value occurring.
In our case:
E(X) = (5 * P(X1)) + (10 * P(X2)) + (15 * P(X3))
Since each interarrival time is equally likely, P(X1) = P(X2) = P(X3) = 1/3.
Replacing the probabilities:
E(X) = (5 * 1/3) + (10 * 1/3) + (15 * 1/3)
Simplifying:
E(X) = (5 + 10 + 15) / 3
E(X) = 30 / 3
E(X) = 10
Therefore, the expected value of the length of the interarrival time is 10 minutes.