3 Mixture of Gaussians [40 points] Let X ,Y be two independent normal RVs, With means M = 100, My = 300 and standard deviations 0,, = 0,, = 10. Consider the RV U dened by 1 2 Alternatively, consider the RV Z that is generated as follows: U = (X + Y). (a) With probability % we sample Z from N (,u, = 100, 02 = 100). (b) With probability % we sample Z from N (,u = 300, a2 = 100). 1. [10 points] Simulate the sampling, and produce two histograms (one for U and one for Z) over 10000 samples for each U, Z. 2. [10 points] Compute the expected values of U, Z. 3. [20 points] Compute the variances of U, Z. Hint: To compute the variance of Z, it will be helpful express Z as a func- tion of X, Y and an indicator variable I that indicates from which of the two distributions (a),(b) the realization of Z was drawn