Let A be a random variable which can be represented as the sum of a group of random variables. A = K1 + K2 + ::: + Kn; (1) where K1;K2; :::;Kn are independent and identically distributed. Therefore, we can write E[K] = E[K1] = E[K2] = ::: = E[Kn]; (2) and V ar(K) = V ar(K1) = V ar(K2) = ::: = V ar(Kn): (3) Find the general expressions for E[A] and V ar[A] by using E[K] and V ar[K].

5 Let K1;K2; :::;Kn be a sequence of Poisson random variables where = 1. By using MATLAB generate 10000 samples of A for each n value in n = f1; 2; 3; 5; 10g. Take the normalized histogram of each sample vector, take the CDF of the histograms, and plot the histogram CDFs separately side by side. (You are encouraged to use subplot here to keep your plots organized.) Calculate the parameters of Gaussian CDF for each case with the help of the expected value and variance expressions you have found earlier and plot them onto their respective histogram CDF plots. How do the plots change with increasing n? Do the CDFs and histograms match? Comment on it. Let us keep n = 10. Generate another 100000 samples of A, take the histogram CDF and plot the Gaussian CDF onto the histogram. What do increasing the number of samples achieve? Comment on it. Let us keep n = 10. Change the parameter of from 1 to 2. Generate 10000 samples and take the histogram CDF. Change the parameters of the Gaussian CDF accordingly and plot it onto the histogram CDF. What changes? Comment on it.