Question

Assume that one can simulate as many \({\bf{i}}.{\bf{i}}.{\bf{d}}.\)exponential random variables with parameters\({\bf{1}}\) as one wishes. Explain how one could use simulation to approximate the mean of the exponential distribution with parameters\({\bf{1}}\).

Short answer

Take the average of \(n\) simulated exponential random variables, which \(n\) is sufficiently large.

Step-by-step solution

Step 1:Definition for the exponential distribution:

• The probability distribution of the time *between* the events in a Poisson process is defined as an exponential distribution.
• When you think about it, the amount of time until the event occurs means that not a single event has occurred during the waiting period.
• In other words, this is Poisson\(\left( {X = 0} \right).\)

To determine the exponential distribution with parameters\({\bf{1}}\):

• The mean can be obtained by simulating random variables from an exponential distribution with parameters\(1\), e.g.\(n = 1000\), and taking their average.
• For example, use command rexp in the programming language\(R\).
• The number of generated random numbers should be large.