[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] be a random variable with cumulative distribution function

</math> F(x) = \left \{ \begin{array}{ll}

                          0, & \mbox{if [math]x  \lt  0[/math]}, \\
            \sin^2(\pi x/2), & \mbox{if [math]0 \leq x \leq 1[/math]},  \\
                          1, & \mbox{if [math]1  \lt  x[/math]}.
               \end{array}
      \right.

[[math]] \ltul\gt\ltli\gt What is the density function $f_X$ for \ltmath\gtX[[/math]]

?

What is the probability that [math]X \lt 1/4[/math]?