Prove that E[X 2 ] ≥ (E[X]) 2 . When do we have equality? Let c be a constant. Show that (a) Var(cX) = c2Var(X); (b) Var(c + X) = Var(X). (a) Calculate E[X] for the maximum random variable of Exercise...

Prove that E[X²] ≥ (E[X])². When do we have equality?

Let c be a constant. Show that

(a) Var(cX) = c2Var(X);

(b) Var(c + X) = Var(X).

(a) Calculate E[X] for the maximum random variable of Exercise 37.

(b) Calculate E[X] for X as in Exercise 33.

(c) Calculate E[X] for X as in Exercise 34.

If X is uniform over (0, 1), calculate E[Xn] and Var(Xn).