3.4 Consider the random variable X with the density f(x) = C cos(x) sin(x), x ∈ [0, π/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 > π/12)

e) Calculate the same probability by integrating the density. Are the two numbers close?

f) Generate 10, 000 numbers and use them to estimate E

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esin(x)

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