Problem

The continuous random variable $X$ has probability density function given by

\dfrac{1}{48}(x^2 - 8x + c) & 2 \leqslant x \leqslant 5 \\ 0 & \text{otherwise} \end{cases}$$ (a) Show that $c = 31$ <div style="text-align: right;">(3)</div> (b) Find $P(2 < X < 3)$ <div style="text-align: right;">(2)</div> (c) State whether the lower quartile of $X$ is less than 3, equal to 3 or greater than 3 Give a reason for your answer. <div style="text-align: right;">(1)</div> Kei does the following to work out the mode of $X$ <div style="text-align: center;">$f'(x) = \dfrac{1}{48}(2x - 8)$</div> <div style="text-align: center;">$0 = \dfrac{1}{48}(2x - 8)$</div> <div style="text-align: center;">$x = 4$</div> <div style="text-align: center;">Hence the mode of $X$ is 4</div> Kei’s answer for the mode is incorrect. (d) Explain why Kei’s method does not give the correct value for the mode. <div style="text-align: right;">(1)</div> (e) Find the mode of $X$ Give a reason for your answer. <div style="text-align: right;">(2)</div> (Total for Question 3 is 9 marks)