Answer the questions below. This requires step by step answers clearly indicating the working. Thanks in advance.

(i) Determine the cumulative distribution function for the random variable having the PDF: f (x) =2pxe Ax' x30 where / is a positive constant. (ii) Hence, derive the PDF of Y = X - if X has the distribution above.Solve the following constrained optimization problem: 1 maximy, subject to: I + exp [-1] + 32 E y, 1: 33 D, and nd optimal (I, y, z}. 1. Consider the following health status problem: maxy/ = - (21-1)- - (62 -2) + 10, subject to : x1 + 12 =1, where y represents an individual's health, and r, and x2 are daily dosages of two health-enhencing drugs. (a) If there is no constraint (2), find optimal (21, x2) and the corresponding maximum value of y.(b) In the presence of constraint (2), find optimal (21, 12, A) (X is the La- grangian multiplier for this constrained optimization problem) and the corre- sponding maximum value of y, and show that A is equal to the reduction of the maximum value of y due to the constraint , +r, = 1