| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Linear combinations of independent variables |
| Difficulty | Challenging +1.8 This is a Further Maths statistics question requiring knowledge that E(XY) = E(X)E(Y) for independent variables (straightforward for part a), but part (b) requires deriving Var(XY) = E(X²)E(Y²) - [E(X)E(Y)]², which involves computing second moments from given means and variances. While the formulas are standard in Further Maths, the multi-step algebraic manipulation and the need to recall/derive the product variance formula makes this significantly harder than typical A-level questions, though not exceptionally difficult for Further Maths students who have studied this topic. |
| Spec | 5.02b Expectation and variance: discrete random variables5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(E(W) = E(X)E(Y) = 168\) | B1 | AO1 |
| (b) \(E(X^2) = (E(X))^2 + \text{Var}(X) = 221\) | M1, A1, A1 | AO1 |
| \(E(Y^2) = 153\) | A1 | AO1 |
| \(\text{Var}(W) = E(W^2) - [E(W)]^2 = E(X^2)E(Y^2) - (E(X)E(Y))^2 = 221 \times 153 - 168^2 = 5589\) | M1, A1, A1 | AO3, AO1, AO1 |
| \(\text{SD} = 74.8\) (74.75961...) | [7] | AO1 |
**(a)** $E(W) = E(X)E(Y) = 168$ | B1 | AO1
**(b)** $E(X^2) = (E(X))^2 + \text{Var}(X) = 221$ | M1, A1, A1 | AO1
$E(Y^2) = 153$ | A1 | AO1
$\text{Var}(W) = E(W^2) - [E(W)]^2 = E(X^2)E(Y^2) - (E(X)E(Y))^2 = 221 \times 153 - 168^2 = 5589$ | M1, A1, A1 | AO3, AO1, AO1
$\text{SD} = 74.8$ (74.75961...) | [7] | AO1
---The random variable $X$ has mean14 and standard deviation 5. The independent random variable $Y$ has mean 12 and standard deviation 3. The random variable $W$ is given by $W = XY$. Find the value of
\begin{enumerate}[label=(\alph*)]
\item E(W), [1]
\item Var(W). [6]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 Q1 [7]}}