Problem(1) Given a continuous random variable X with probability density function

fX (x) = f(x), say, x can be any real number.

(a) What is the definition of the expectation of the random variable X, i.e. E(X) =?

(b) If we denote E(X) by X , then the definition of the variance of the random

variable X is defined by Var(X) = E[(X X )2]; show that Var(X) = E(X2) 2

X .

(c) If we have another continuous random variable Y (with expectation Y ), the covariance of the two random variables X and Y is defined by Cov(X, Y ) = E[(X X )(Y Y )] . Show that Cov(X, Y ) = E(XY ) E(X)E(Y ). (That is, Cov(X, Y ) = E(XY ) X Y .)