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.
11.67
Well, if an observer arrives at some particular time, unrelated to the history of the process, they won't have any information about what happened before. So, they're in a bit of a pickle, aren't they? It's like trying to predict the future without a crystal ball or a time-traveling flamingo.
But fear not! We can still crunch some numbers and find the expected value of the length of the interarrival time for our befuddled observer. Since the interarrival times are equally likely to be 5, 10, or 15 minutes, we can find the expected value by taking the average of these values.
Let's do some math, shall we? (Don't worry, it won't hurt... much.)
The expected value (E) of a random variable X with possible outcomes x₁, x₂, ..., xn, each with respective probabilities p₁, p₂, ..., pn, is given by:
E(X) = x₁ * p₁ + x₂ * p₂ + ... + xn * pn
In our case, we have three possible interarrival times: 5, 10, and 15 minutes, each with equal probabilities of 1/3. So, we can plug these values into the equation:
E(X) = 5 * (1/3) + 10 * (1/3) + 15 * (1/3)
= 5/3 + 10/3 + 15/3
= (5 + 10 + 15) / 3
= 30 / 3
= 10
Hence, the expected value of the length of the interarrival time seen by our confused observer is 10 minutes. But hey, who needs precision when you can have confusion, right? Enjoy your time calculations, my friend!
To find the expected value of the length of the interarrival time, we need to calculate the average value of all possible interarrival times.
Let's denote the random variable representing the interarrival time as X, and let's denote the probabilities of the interarrival times being 5, 10, and 15 minutes as P(X = 5), P(X = 10), and P(X = 15), respectively.
Since the interarrival times are i.i.d. random variables, we can calculate the expected value (or mean) using the formula:
E(X) = (5 * P(X = 5)) + (10 * P(X = 10)) + (15 * P(X = 15))
In this case, since the interarrival times are equally likely to be 5, 10, or 15 minutes, we have:
P(X = 5) = P(X = 10) = P(X = 15) = 1/3
Substituting these probabilities into the formula, we get:
E(X) = (5 * 1/3) + (10 * 1/3) + (15 * 1/3)
E(X) = (5/3) + (10/3) + (15/3)
E(X) = 30/3
E(X) = 10 minutes
Therefore, the expected value of the length of the interarrival time seen by an observer who arrives at some particular time is 10 minutes.
To find the expected value of the length of the interarrival time, we need to calculate the average value of the interarrival time.
Given that the consecutive interarrival times are identically and independently distributed (i.i.d.) random variables, and equally likely to be 5, 10, or 15 minutes, we can use the formula for the expected value (E) of a discrete random variable:
E[X] = Σ(x * P(X = x))
Where X is the random variable, x is the value of the random variable, and P(X = x) is the probability of X taking on the value x.
In this case, the random variable is the length of the interarrival time, denoted by T.
Let's calculate the expected value step-by-step:
1. Calculate the probability of each possible interarrival time:
P(T = 5) = 1/3
P(T = 10) = 1/3
P(T = 15) = 1/3
2. Multiply each interarrival time by its respective probability:
(5 * 1/3) + (10 * 1/3) + (15 * 1/3) = (5/3) + (10/3) + (15/3) = 30/3 = 10
So, the expected value of the length of the interarrival time is 10 minutes.
In conclusion, an observer who arrives at some particular time, unrelated to the history of the process, can expect, on average, an interarrival time of 10 minutes.