Problem

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

Given that

• $P(2 < X < 5) = \dfrac{3}{16}$
• $E(X) = 6$

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

(b) find the value of the constant $c$ such that $E(cX + 1) = 3$

(c) find the exact value of $\text{Var}(X)$

(d) find the exact value of $E(X^2)$

(e) find $P\left(\dfrac{3}{2}X - b > a\right)$

(Total for Question 6 is 10 marks)