Answer:

- The density function \(f_X\) for X can be computed by taking the derivative of the

cumulative distribution function \(F(X)\)

The derivative of a constant is 0 so we will only differentiate the function

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

We could use the chain rule to differentiate the function or use a trignometric identity:

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

to make it easier.

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

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

- 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

\(\int_{0}^{1/4} sin^2(\pi x/2) dx\)

we use the trig identity again to make integrating the function easier

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

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