2. The amount of flour used by a factory in a week is \(Y\) thousand kg where \(Y\) has probability density function
$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c }
k \left( 4 - y ^ { 2 } \right) & 0 \leqslant y \leqslant 2
0 & \text { otherwise }
\end{array} \right.$$
- Show that the value of \(k\) is \(\frac { 3 } { 16 }\)
Use algebraic integration to find
- the mean number of kilograms of flour used by the factory in a week,
- the standard deviation of the number of kilograms of flour used by the factory in a week,
- the probability that more than 1500 kg of flour will be used by the factory next week.