Assume that a source of U(0, 1) random numbers U1, U2, . . . is readily available. Consider the following probability density function of the continuous random variable X:

f(x) = (1/12)(x + 2)^3 , if 2 x < 0

(2/27)(3 x)^2 , if 0 x 3

0 , otherwise.

(a) Determine the cumulative density function F(x) of X for all < x < .

(b) Construct an algorithm for generating independent and identically distributed random variates for X by the composition method. Show all steps explicitly and clearly.

(c) Assume that the following uniform random numbers are available:

0.6, 0.9, 0.7, 0.3, 0.5, 0.1.

Generate the first X (correct to 3 significant figures) by your algorithm.