Problem

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

Given that

• $P(X > 27) = \dfrac{3}{4}$
• $\text{Var}(X) = 300$

(a) find the value of $a$ and the value of $b$

Given also that

$4 \times P(X < k - 10) = P(X > k + 20)$

(b) find the value of $k$

(ii) A piece of wire of length 42 cm is cut into 2 pieces at a random point.

Each of the two pieces of the wire is bent to form the outline of a square.

Find the probability that the side length of the larger square minus the side length of the smaller square will be greater than 2 cm.

(Total for Question 4 is 9 marks)