Let X and Y be two independent random variables, both uniformly distributed on the interval (0, 1) (so they are independent and have the same distribution).

a) Compute the cumulative distribution function (cdf) and the density of the random variable Z = max{X, Y }.

b) Compute the expectation and the standard deviation of Z.

Additional note: X and Y are two numbers picked independently and uniformly from the interval (0, 1) and Z is the maximum of the two. The cumulative distribution function of Z is