10 The continuous random variable \(X\) has probability density function given by
\(f ( x ) = \begin{cases} \frac { 4 } { 15 } \left( \frac { a } { x ^ { 2 } } + 3 x ^ { 2 } - \frac { 7 } { 2 } \right) & 1 \leqslant x \leqslant 2 ,
0 & \text { otherwise, } \end{cases}\)
where \(a\) is a positive constant.
- Find the cumulative distribution function of \(X\) in terms of \(a\).
- Hence or otherwise determine the value of \(a\).
- Show that the median value \(m\) of \(X\) satisfies the equation
$$8 m ^ { 4 } - 28 m ^ { 2 } + 9 m - 4 = 0 .$$
- Verify that the median value of \(X\) is 1.74, correct to \(\mathbf { 2 }\) decimal places.
- Find \(\mathrm { E } ( X )\).
- Determine the mode of \(X\).