Question: Prove that adding 12 i.i.d. RVs each on U(0,1) yields a RV Z, where \(Z=\sum\limits_{i=1}^{12}{{{X}_{i}}}-6\) where Xi~U(0,1) such that Z has a mean of 0 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 with the same mean and standard deviation (hint: the errors occur at the tails). Explain why there is error. Show how your random number generator can produce any Gaussian distribution with an arbitrary mean and variance by simply by using the identity \(Z=\frac{X-\mu }{\sigma }\)