Suppose that a continuous variable X has density function f (x) =

{

cxs, 0 ≤ x ≤ a, 0 elsewhere, where a and s are two given positive real numbers, while c is a suitable constant.

(i) Find the value of c in terms of a and s.

(ii) Obtain the distribution function for each of the following random variables:

X1 = eX, X2 = X2, X3 = (X − 1)2, X4 =

√

X, X5 = 1∕X.

(iii) Calculate the mean and variance for each of the variables X2, X3, X4, X5 in Part (ii).