Assume that Y and X are random variables, while a and b are constant parame- ters. Recall that the variances of Y and X are defined as: var(Y ) = E (Y E(Y ))2 and var(X) = E (X E(X))2, and the covariance between the two variables is defined as: cov(Y, X) = E [(Y E(Y ))(X E(X))].

(a) Assume that Y = a + bX. Use the definitions of var(Y ) and var(X) to show that var(Y ) = b2var(X).

(b) Assume again that Y = a + bX. Use the definitions of cov(Y, X) and var(X) to show that cov(Y, X) = b var(X).

(c) Use the results from the previous two points to show that if Y = a + bX, then corr(Y, X) = cov(Y,X) is equal to either +1 or 1 depending on the sign of the parameter b.

var(Y )var(X)