As shown in the diagram, the continuous random variable \(X\) has probability density function f given by
\(f(x)=\left\{\begin{array}{ll} a & 0 \leqslant x \leqslant 5 \\ b-c x & 5 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{array}\right.\)
where \(a, b\) and \(c\) are constants.
(a) Show that \(a=\frac{2}{13}\) and find the values of \(b\) and \(c\).

(b) Find the mean of \(X\).

(c) Find the median of \(X\).

The random variable \(Y\) is defined by \(Y=X^{2}\).
(d) Find the cumulative distribution function for \(Y\).