2026 MAY TZA P2 Q7

The continuous random variable \(x\) has a probability distribution \(f\) defined as

\(f(x) = \begin{cases} \frac{b}{a}x, & 0 \leq x \leq a \\ \frac{b}{a - 5}(x - 5), & a < x \leq 5 \end{cases}\) where \(0 < a < 5\) and \(b \in \mathbb{Q}^+\)

The graph of \(f\) is shown in the following diagram.

1. Show that \(b = 0.4\).

   [2]

It is given that \(\mathrm{E}(X) = 3.2\).

2. Find the value of \(a\).

   [4]