- The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 18 } ( 11 - 2 x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
- Find \(\mathrm { P } ( \mathrm { X } < 3 )\)
- State, giving a reason, whether the upper quartile of \(X\) is greater than 3, less than 3 or equal to 3
Given that \(\mathrm { E } ( \mathrm { X } ) = \frac { 9 } { 4 }\)
- use algebraic integration to find \(\operatorname { Var } ( \mathrm { X } )\)
The cumulative distribution function of \(X\) is given by
$$F ( x ) = \left\{ \begin{array} { l r }
0 & x < 1
\frac { 1 } { 18 } \left( 11 x - x ^ { 2 } + c \right) & 1 \leqslant x \leqslant 4
1 & x > 4
\end{array} \right.$$ - Show that \(\mathrm { c } = - 10\)
- Find the median of \(X\), giving your answer to 3 significant figures.
\section*{Q uestion 2 continued}