| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2023 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Apply E(aX+b) or Var(aX+b) formulas directly |
| Difficulty | Easy -1.2 This is a straightforward application of the linear transformation formula E(aX+b) = aE(X)+b with a simple discrete distribution. Students only need to calculate E(X) from the given probabilities (a routine calculation) and then apply the formula directly. This requires basic recall and arithmetic with no problem-solving or conceptual challenge. |
| Spec | 5.02b Expectation and variance: discrete random variables |
| \(x\) | - 4 | 3 | 8 |
| \(\mathrm { P } ( X = x )\) | 0.2 | 0.7 | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = -4 \times 0.2 + 3 \times 0.7 + 8 \times 0.1 = 2.1\) | M1 | Uses correct formula for the mean of a discrete random variable to calculate \(E(X)\) or \(E(5X)\), or calculates values of \(5x-7\) for each value of \(x\) |
| \(E(5X-7) = 5E(X) - 7\) | M1 | States or uses correct formula for \(E(5X-7)\). Not implied by sight of \(E(5X-7) = 3.5\) following a correct value of \(E(X)\) or \(E(5X)\) |
| \(E(5X-7) = 5 \times 2.1 - 7 = 3.5\) | R1 | Completes a reasoned argument by giving a calculation that obtains the given value of \(E(5X-7)\) |
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = -4 \times 0.2 + 3 \times 0.7 + 8 \times 0.1 = 2.1$ | M1 | Uses correct formula for the mean of a discrete random variable to calculate $E(X)$ or $E(5X)$, or calculates values of $5x-7$ for each value of $x$ |
| $E(5X-7) = 5E(X) - 7$ | M1 | States or uses correct formula for $E(5X-7)$. **Not** implied by sight of $E(5X-7) = 3.5$ following a correct value of $E(X)$ or $E(5X)$ |
| $E(5X-7) = 5 \times 2.1 - 7 = 3.5$ | R1 | Completes a reasoned argument by giving a calculation that obtains the given value of $E(5X-7)$ |
**Question total: 3 marks**3 The discrete random variable $X$ has probability distribution
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$x$ & - 4 & 3 & 8 \\
\hline
$\mathrm { P } ( X = x )$ & 0.2 & 0.7 & 0.1 \\
\hline
\end{tabular}
\end{center}
Show that $\mathrm { E } ( 5 X - 7 ) = 3.5$\\
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2023 Q3 [3]}}