Often times the probability distribution of a Random Variable of interest is unknown. In such cases simulation can be a useful tool to estimate the probability distribution. Suppose that X1, X2, X3 are each Poisson(2 = 5) independent random variables. And let Y = maximum(X1, X2, X3). We are interested in the probability distribution of Y. Note: Y is a discrete RV. Randomly generate N = 1000000 values for each of X1, X2, X3. Matlab code: >>N=1000000; >>lambda=5; >>Xl=poissind(lambda, [N,1]); % repeat these steps for X2 and X3. To create vector Y where Y = maximum(X1, X2, X3) we can use: >>Y=max(X1,max(X2,X3)); % Produces vector: Y ) vector: Y = [y] = [max (x1, x2, x36] = = Note that Y is a discrete RV (possible values 0,1,2,3, ...). We estimate Py(y) = P(Y = y) by the proportion of times Y = y. Create a labelled, normalized histogram of Y. "Normalized" for a discrete random variable means to express the column height as proportions (Thus, across all values of Y, the proportions must sum to) For a discrete RV, you must be careful that the number of bins (i.e. columns on your graph) aligns with the integer values in your data set. You want 1 integer in each bin. If these are not aligned, you'll see gaps or weird spikes on your graph. You may have to try a couple until you get something you like. A promising candidate is: >>bins=max(Y)-min(Y)+1; To create the labelled, normalized histogram you can use: >> histogram(Y,bins, 'normalization’, probability') >>title('Maximum of 3 Independent Poisson Random Variables ) >>xlabel((Y) >>ylabel('Estimated PMF of Y') Note: To normalize a discrete RV (as in this case) use 'probability'. To normalize a continuous RV (as you did in a previous Homework problem) use ‘pdf". In addition, compute the mean and standard deviation of Y; the commands are: mean(Y) and std(Y) respectively. Note these would be denoted 7 and sy respectively since they are based only on our sample results they are estimates of ly and Oy respectively. For you to hand in: a. labelled, normalized histogram of Y b. mean and standard deviation of vector Y c. Use your histogram results to estimate P(Y <5). >>Prob=(sum(Y<=5)/N) d. Include your MatLab code.