Consider an arrival process whose inter arrival times are independent exponential random variables with mean 2 (and consequently variance equal to 4) and consider the inter arrival interval S seen by an observer who arrives at a fixed time t^*, as in the preceding video. What is the variance of S?
To find the variance of the interarrival interval S, we can use the properties of exponential random variables.
The variance of an exponential random variable with mean μ is given by σ^2 = μ^2, where σ is the standard deviation.
In this case, the mean interarrival time is 2, so the variance of each exponential random variable is (2^2) = 4.
Since the interarrival times are independent, the variance of the sum of n exponential random variables is equal to the sum of the variances of each individual variable.
Since S is the sum of multiple interarrival times, each with variance 4, the variance of S is given by:
Variance(S) = n * Variance(interarrival time) = n * 4,
where n is the number of interarrival times.
Note that the actual value of n depends on the specific context or time frame being considered.
The variance of the interarrival interval S can be calculated using the property of exponential distributions, which states that the variance of an exponential random variable is equal to the square of its mean.
Given that the mean of the interarrival times is 2, the variance of each interarrival time is 4. Since S is the sum of n independent exponential random variables, each with variance 4, the variance of S is equal to n times the variance of each individual exponential random variable.
In this case, n represents the number of interarrival times that occur between the observer's arrival time t^* and the subsequent arrival time.
Therefore, the variance of S is n * 4.