Chapter 5 (Distributions and Densities) - Exercise 17 page: 221

a) The density function \(f_X\) for X is nothing but the derivative of the cumulative distribution function \(F(X)\)

\(d/dx(F(X)) = d/dx(sin^2(\pi x/2))\) for \(0 \le x \le 1\)

We know that:

\(sin^2(\theta) = (1-cos(2\theta)) / 2\)

so:

\(d/dx(sin^2(\pi x/2)) = d/dx((1-cos(2 \pi x/2)) / 2) = d/dx(1/2 - cos(2(\pi x/2))/2) = d/dx\big(1/2 - cos(\pi x) / 2\big)\)

Therefore: \(f_X = \pi sin(\pi x)/2\)

b) Finding the probably we have to use the cumulative distribution function F(X) and compute integrals.

For the probability that X < 1/4, we compute \(\int_{0}^{1/4} sin^2(\pi x/2) dx\)

Let’s intergrate:

\(1/2\int_{0}^{1/4} (1 - cos(\pi x)) dx = 1/2 \bigg[x - sin(\pi x)/ \pi \bigg]_{0}^{1/4} =\)

\(1/2\bigg(1/4 - sin(\pi/4)/ \pi\bigg) = 1/8 - \sqrt{2}/4 \pi\)