Consider the random variable X with the density: f (x) = C cos(x) sin(x), x [0, pi/2].

(a) Calculate the constant C which makes above a probability density.

(b) Sketch this density.

(c) Implement the importance sampling method to generate random numbers from the density. Create a histogram by generating 1000 such numbers and compare with the previous part.

(d) Generate 1000 such numbers and use them to estimate the probability: P(X > /12). Exercises 209

(e) Calculate the same probability by integrating the density. Are the two numbers close? (f) Generate 10, 000 and use them to estimate E [e^ sin(x) ]

Must use R programming with code