poissonProcess = function(tot, n,lambda, s, nmbSim) { # initiating vector where I will store all of my estimates vectorOfFractions = c(rep(0,nmbSim)) # Simulating the Poisson process nmbSim times for (j in 1:nmbSim) { # simulating little t values (incremental arrival times of customers) tVec = rexp(tot,lambda); # calculating big T values (summing up the times) TVec = cumsum(tVec) # Calculating the fraction of customers that arrive before time s fracLessThenS = length(TVec[TVec < s]) # Storing it in 'vectorOfFractions' if (fracLessThenS == n) { vectorOfFractions[j] = 1; } } estProb = mean(vectorOfFractions); paste("estimated Probability = ", estProb); }

1.In the program PoissonProcess.R, you can simulate a Poisson process from scratch. It allows you to do the following nmbSim times (nmbSim is the total number of simulations): simulate tot exponentials with parameter and count the fraction of times n customers arrive before time s. As youve learned in this module, the number of occurrences before time s is distributed as a Poisson random variable. Run the program Ive written for you, and use the function dpois in R to illustrate that the number occurrences before time s does, in fact, follow a Poisson random variable.

2.show how the accuracy of the Poisson approximation varies when the maximum of the arrival probabilities.