1. Generate 1000 Poisson random numbers with )1 = 1.9, and plot a bar chart of the resulting sample . a=rpois (1000 , lambda =1.9) barplot (table (a)) Calculate the mean and variance of your sample. Compare with their theoretical values. 2. Generate 200 Poisson random numbers with /\\ = 2.7, and plot a bar chart of the resulting sample . Use plot{dpois(x=0:20, lambda=2.7), type="b") to plot theoretical probabilities up to a: = 20 and compare with your barchart. Using your sample to estimate the mean ofX, where X N Pois(2.7). It can be shown that the theoretical value of E(X3} is /\\+3/\\2+/\\3. Compare this theoretical mean with your sample estimate 3. Q4.2.12 : Midwestern Skies books ten commuter ights each week. Passenger totals are much the same from week to week, as are the numbers of pieces of luggage that are checked. Listed in the following table are the numbers of bags that were lost during each of the rst forty weeks in 2009 . Do these Engines support the presumption that the number of bags lost by Midwestern during a typical week is a Poisson random variable? Week Bags Lost Week Bags Lost Week Bags Lost 1 1 14 2 2'? 1 2 f] 15 1 28 2 3 0 16 3 29 0 4 3 17 0 30 0 5 4 18 2 31 1 6 1 19 5 32 3 7 U 20 2 33 1 8 2 21 1 34 2 9 0 22 1 35 0 10 2 23 1 36 1 11 3 24 2 37 4 12 1 25 1 38 2 13 2 26 3 39 1 40 0 Estimate A for a suitable Poisson model for these data and complete the table below