question 1

Set the seed to be 123. Simulate U 1, U 2, U 3(each with size 10,000) all come from a uniform(0,1) distribution. (a) Caculate M = max(U 1, U 2, U 3) using the simulated observations, and visualize the rst 6 values. (b) Plot

M. (c) Estimate (to 3 signicant digits) the probability P(M > 0.75) (Hint, use the number of elements in M

greater than 0.75 to be divided by the total number of values).

question 2

Set the seed to be 123. Similate 3,000 points from a mixture of two normal distributions. Normal distribution 1 is the standard normal. Normal distribution 2 has mean 1 and the variance twice the size of the distribution 1. Moreover, sample 1 from distribution 1 should be twice the size of sample 2 from distribution 2. Plot the histogram of sample 1 (from distribution 1), sample 2 (from distribution 2), and the combind sample (from the mixture) on the same axes.