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 seen by an observer who arrives at some particular time, we need to find the average value considering all possible interarrival time scenarios.
Given that the consecutive interarrival times are independent and identically distributed (i.i.d.) random variables, we can calculate the expected value using the following formula:
Expected Value = Σ (x * P(x))
Where:
- x represents each possible interarrival time.
- P(x) represents the probability of each interarrival time occurring.
In this case, the possible interarrival times are 5 minutes, 10 minutes, and 15 minutes, and they are equally likely to occur.
Let's calculate the expected value step by step:
Step 1: Determine the probabilities of each interarrival time occurring.
Since the interarrival times are equally likely, the probability of each interarrival time is 1/3. Therefore:
- P(5 minutes) = 1/3
- P(10 minutes) = 1/3
- P(15 minutes) = 1/3
Step 2: Calculate the expected value.
Expected Value = (5 minutes * P(5 minutes)) + (10 minutes * P(10 minutes)) + (15 minutes * P(15 minutes))
Expected Value = (5/3) + (10/3) + (15/3)
Expected Value = 30/3
Expected Value = 10 minutes
Hence, the expected value of the length of the interarrival time seen by an observer who arrives at some particular time is 10 minutes.