Problem

(i) The continuous random variable $X$ is uniformly distributed over the interval $[a, b]$

Given that $P(5 < X < 13) = \dfrac{1}{5}$ and $E(X) = 9$, find $P(3X > a + b)$

(ii) The continuous random variable $Y$ is uniformly distributed over the interval $[1, c]$

Given that $\operatorname{Var}(Y) = 0.48$, find the exact value of $E(Y^2)$

(iii) A wire of length 20 cm is cut into 2 pieces at a random point. The longest piece of wire is then cut into 2 pieces, equal in length, giving 3 pieces of wire altogether.

Find the probability that the length of the shortest piece of wire is less than 6 cm.

(Total for Question 7 is 12 marks)