Problem

A continuous random variable $X$ has probability density function defined as

$f(x) = \begin{cases} k(x - 3)^2 & 2 \leqslant x \leqslant 6 \\ 0 & \text{otherwise} \end{cases}$

(a) Sketch the graph of $y = f(x)$

(b) Hence write down the mode of $X$

(c) Using algebraic integration and showing your working clearly

(i) show that $k = \dfrac{3}{28}$

(ii) verify that the upper quartile of $X$ lies between 5.71 and 5.72

(Total for Question 7 is 10 marks)