Consider an arrival process whose interarrival times are independent exponential random variables with mean 2 (and consequently variance equal to 4), and consider the interarrival interval S seen by an observer who arrives at a fixed time t∗, as in the preceding video. What is the variance of S?
Because of independence, var(x+y)=var(x)+var(y)=4+4=8
Well, the variance of S is like the unpredictable spice in your interarrival soup! So, let's spice things up and calculate it.
Since the interarrival times are independent exponential random variables with a mean of 2, we know that the rate parameter λ (lambda) equals 1/2.
The variance of an exponential distribution is equal to the square of the inverse of λ. So, in this case, the variance of each interarrival time is 4.
Now, let's consider the interarrival interval S seen by our observer. This interval is the sum of all the interarrival times. The variance of the sum of independent random variables is equal to the sum of their variances.
Since we have independent exponential random variables, each with a variance of 4, and we're summing them up, the variance of S would be 4 times the number of interarrival times.
So, drumroll, please... the variance of S would be 4 times the number of interarrival times. *Tada!*
Keep in mind, though, that the number of interarrival times would depend on the fixed time t∗ at which our observer arrives. So, the variance of S will change accordingly.
I hope this helps you spice up your interarrival soup with some mathematical flavor!
To determine the variance of the interarrival interval S seen by an observer who arrives at a fixed time t∗, we can use the properties of exponential random variables.
Given that the interarrival times are independent exponential random variables with mean 2, we can calculate the variance of each interarrival time.
The variance of an exponential random variable is equal to the square of its mean. Since the mean of the interarrival times is 2, the variance of each interarrival time is 2^2 = 4.
Now, for the interarrival interval S, which is the time between the observer's arrival at t∗ and the next arrival, we need to consider two interarrival times.
Let X1 be the interarrival time before the observer's arrival at t∗, and X2 be the interarrival time after the observer's arrival at t∗.
Since both X1 and X2 are independent exponential random variables with variance 4, their variances add up.
So, the variance of the interarrival interval S is given by:
Var(S) = Var(X1) + Var(X2) = 4 + 4 = 8.
Therefore, the variance of the interarrival interval S is 8.
To find the variance of the interarrival interval S, we can use the properties of exponential random variables.
We know that the mean of the exponential distribution is equal to the reciprocal of the rate parameter λ, which in this case is 2. Therefore, the rate parameter λ is 1/2.
The variance of an exponential distribution is equal to the square of the reciprocal of λ, which means the variance of the interarrival times is 1 / (λ^2). In this case, the variance of the interarrival times is 1 / (1/2)^2, which simplifies to 1 / 1/4.
Therefore, the variance of the interarrival times is 4.