Problem

The continuous random variable $X$ has probability density function $f(x)$ given by

$f(x) = \begin{cases} \dfrac{1}{20}x^3 & 0 \leqslant x \leqslant 2 \\[6pt] \dfrac{1}{10}(6 - x) & 2 < x \leqslant 6 \\[6pt] 0 & \text{otherwise} \end{cases}$

(a) Sketch the graph of $f(x)$ for all values of $x$

(b) Write down the mode of $X$

(c) Show that $P(X > 2) = 0.8$

(d) Define fully the cumulative distribution function $F(x)$

Given that $P(X < a \mid X > 2) = \dfrac{5}{8}$

(e) find the value of $F(a)$

(f) Hence, or otherwise, find the value of $a$. Give your answer to 3 significant figures.

(Total for Question 6 is 15 marks)