The density function of a variable X is given by f (x) =

{

24∕x4, x ≥ 2 0, x < 2.

(i) Find the mean and variance of X.
(ii) Obtain the distribution function, F, of X and use it to calculate the conditional probability P(X ≤ 7|X ≥ 4).
(iii) Find the probability density, the mean and the variance of the random variable Y = 1∕X.
(iv) Which, if any, of the following five equations are true for the variables X and Y defined above?
E(1∕X) = 1∕E(X), Var(1∕X) = 1∕Var(X), E(X + Y) = E(X) + E(Y), E(X + Y−1) = 2E(X), E(XY) = E(X)E(Y).