7. A corner-shop has weekly sales (in thousands of pounds), which can be modelled by the continuous random variable \(X\) with probability density function
$$\begin{array} { l l }
\mathrm { f } ( x ) = k ( x - 2 ) ( 10 - x ) & 2 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } .
\end{array}$$
- Show that \(k = \frac { 3 } { 256 }\) and write down the mean of \(X\).
- Find the standard deviation of the weekly sales.
- Find the probability that the sales exceed \(\pounds 8000\) in any particular week.
If the sales exceed \(\pounds 8000\) per week for 4 consecutive weeks, the manager gets a bonus.
- Find the probability that the manager gets a bonus in February.