| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Linear combinations of independent variables |
| Difficulty | Moderate -0.8 This is a straightforward application of standard expectation and variance rules for linear combinations of independent random variables. Part (a) and (b) require only direct formula application (E[aX+bY+c] and Var[aX+bY+c]), while part (c) uses independence to write E[X²Y]=E[X²]E[Y], then applies Var(X)=E[X²]-E[X]². All three parts are routine calculations with no problem-solving or conceptual challenge, making this easier than average even for Further Maths. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(4Y - 2X + 1) = 4E(Y) - 2E(X) + 1 = 4 \times 10 - 2 \times 17 + 1 = 7\) | M1 A1 | Use of |
| \(\text{Var}(4Y - 5X + 3) = 4^2\text{Var}(Y) + 5^2\text{Var}(X) = 16 \times 16 + 25 \times 64 = 1856\) | M1 A1 | cao |
| \(E(X^2) = \text{Var}(X) + (E(X))^2 = 64 + 17^2 = 353\) | M1 | Use of |
| \(E(X^2Y) = E(X^2) \times E(Y)\) \(E(X^2Y) = 353 \times 10 = 3530\) | M1 A1 | FT their \(E(X^2)\) provided \(\neq 17^2\); cao |
| Total [7] |
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(4Y - 2X + 1) = 4E(Y) - 2E(X) + 1 = 4 \times 10 - 2 \times 17 + 1 = 7$ | M1 A1 | Use of |
| $\text{Var}(4Y - 5X + 3) = 4^2\text{Var}(Y) + 5^2\text{Var}(X) = 16 \times 16 + 25 \times 64 = 1856$ | M1 A1 | cao |
| $E(X^2) = \text{Var}(X) + (E(X))^2 = 64 + 17^2 = 353$ | M1 | Use of |
| $E(X^2Y) = E(X^2) \times E(Y)$ $E(X^2Y) = 353 \times 10 = 3530$ | M1 A1 | FT their $E(X^2)$ provided $\neq 17^2$; cao |
| **Total [7]** | | |The random variable $X$ has mean 17 and variance 64. The independent random variable $Y$ has mean 10 and variance 16. Find the value of
\begin{enumerate}[label=(\alph*)]
\item E$(4Y - 2X + 1)$, [2]
\item Var$(4Y - 5X + 3)$, [2]
\item E$(X^2 Y)$. [3]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2023 Q1 [7]}}