- The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 1 } { 48 } \left( x ^ { 2 } - 8 x + c \right) & 2 \leqslant x \leqslant 5
0 & \text { otherwise }
\end{array} \right.$$
- Show that \(c = 31\)
- Find \(\mathrm { P } ( 2 < X < 3 )\)
- State whether the lower quartile of \(X\) is less than 3, equal to 3 or greater than 3 Give a reason for your answer.
Kei does the following to work out the mode of \(X\)
$$\begin{aligned}
f ^ { \prime } ( x ) & = \frac { 1 } { 48 } ( 2 x - 8 )
0 & = \frac { 1 } { 48 } ( 2 x - 8 )
x & = 4
\end{aligned}$$
Hence the mode of \(X\) is 4
Kei's answer for the mode is incorrect. - Explain why Kei's method does not give the correct value for the mode.
- Find the mode of \(X\)
Give a reason for your answer.