7. A continuous random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { x ^ { 2 } } { 312 } & 4 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the variance of \(X\).
3. Find the cumulative distribution function \(\mathrm { F } ( x )\), for all values of \(x\).
4. Hence find the median value of \(X\).
5. Write down the modal value of \(X\). It is sometimes suggested that, for most distributions, $$2 \times ( \text { median } - \text { mean } ) \approx \text { mode } - \text { median } .$$
6. Show that this result is not satisfied in this case, and suggest a reason why.