Question: Prove theoretically from analysis that adding together 12 i.i.d. uniform RVs on U(-1/2 , 1/2) together yields a RV Z, where \(Z=\sum\limits_{i=1}^{12}{{{X}_{i}}}\) and where Xi~U(-1/2,1/2) such that Z has mean of zero and standard deviation of 1. Plot histograms of thousands of runs of Z (samples of Z) by using a random number generator in your favorite simulator (such as MATLAB), and find out where your random number generator approach is more than 1% off of a true Gaussian distribution of mean of zero and standard deviation of 1 (hint: the errors occur at the tails). Explain why there is error, and how it could be improved.