The random variable X has probability density function k3: forO S a: g 1, _ k for1

The random variable X has probability density function k fx@) forl < 2, k(3 rx) for 2 < c 3, otherwise. (a) Find k so that fx@) is a probability density function. (b) Derive the cumulative distribution function Fx@). (c) Sketch the graphs of fx@) and Fx@). (d) Calculate P[1.5 < X 2.5]. (5 marks) (5 marks) (5 marks) (5 marks) You may use a geometrical method or a calculus method to evaluate the relevant part of this problem. The formula f Cdrx + c may also be used. 2