Problem. In this assignment, you will examine the convergence of sample mean. (You are allowed to use existing functions for computing distributions and generating samples such as binocdf and binornd.) Let X be the Bernoulli(p = 1/2) random variable. Its sample mean Mn(X) = (X1+ X2+ . . . + Xn) is defined using IID Bernoulli(p) random variables X1, X2, ..., Xn. A sequence of sample mean values of Mk (X ) for k = 1, 2, . .., n is called a sample mean trace. (a) Generate m = 1000 sample mean traces (each of length n = 100). At each step k = 1, 2, ..., n, compute the number mk of traces that is within one standard error ek = VVar(Mk (X) ) of the mean E[Mk(X)]. Plot qk = mk/m as a function of k. (b) Using the binomial CDF, derive a closed-form expression of the exact probability pr of the event that Mk(X) is within one standard error of the mean, that is, PK = P(|Mk(X) - E[MK(X)]| x}.) (c) Plot PK as a function of k for k = 1, 2, ..., n. (d) Derive a closed-form expression of limn- P(|Mn(X) - E[Mn(X)]|