Hello I need your help please about Variance

1- Can we have a random variable X with E(X) = 3 and E(X2) = 8?

2- Let X be a random variable with density function f (x) = 1 + x for 1 x 0, f(x) = 1 x for 0 x 1, and f(x) = 0 elsewhere. (a) Justify geometrically that E(X) = 0 (without calculating it). (b) Find Var(X).

3- (Bonus). Suppose X1, . . . , Xn are independent with E (Xi) = and Var(Xi) = 2 for all i = 1,...,n . Find E [(X1 + ... + Xn)2]. (Hint: Use that E(Y2) = Var(Y) + (E (Y))2.)

4- Show that E((X c )2) is minimized by taking c = E(X). (Hint: Use Calculus to study the function f(c) = E ((X c )2 ).)