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Chapter 12: Q12.1-1E (page 791)

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

Expert verified

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

Step by step solution

01

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\(\left( {X = 0} \right).\)
02

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.

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