| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Multiple unknowns from expectation and variance |
| Difficulty | Standard +0.3 This is a standard Further Maths statistics problem requiring three simultaneous equations from probability axioms, expectation, and variance definitions. While it involves algebraic manipulation and substitution across multiple equations, the approach is methodical and follows directly from standard formulas without requiring novel insight or complex problem-solving strategies. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(\boldsymbol { r }\) | - 2 | 0 | \(a\) | 4 |
| \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\) | 0.3 | \(b\) | \(c\) | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.4 + b + c = 1 \Rightarrow b + c = 0.6\) | M1 | Uses sum of probabilities \(= 1\) |
| \(E(R) = 0.2 \Rightarrow (-2 \times 0.3) + (0 \times b) + (a \times c) + (4 \times 0.1) = 0.2 \Rightarrow ac = 0.4\) | M1 | Uses formula for \(E(R)\) |
| \(E(X^2) - (E(X))^2 = (4 \times 0.3) + (0 \times b) + (a^2 \times c) + (16 \times 0.1) - (0.2)^2 = 3.56 \Rightarrow a^2c = 0.8\) | M1 | Uses formula for variance \(E(X^2) - (E(X))^2\) |
| From (2) and (3): \(a = 2\), hence \(c = 0.2\) and \(b = 0.4\) | A1 | Obtains \(a\), \(b\) and \(c\); CAO |
## Question 3:
$0.4 + b + c = 1 \Rightarrow b + c = 0.6$ | M1 | Uses sum of probabilities $= 1$
$E(R) = 0.2 \Rightarrow (-2 \times 0.3) + (0 \times b) + (a \times c) + (4 \times 0.1) = 0.2 \Rightarrow ac = 0.4$ | M1 | Uses formula for $E(R)$
$E(X^2) - (E(X))^2 = (4 \times 0.3) + (0 \times b) + (a^2 \times c) + (16 \times 0.1) - (0.2)^2 = 3.56 \Rightarrow a^2c = 0.8$ | M1 | Uses formula for variance $E(X^2) - (E(X))^2$
From (2) and (3): $a = 2$, hence $c = 0.2$ and $b = 0.4$ | A1 | Obtains $a$, $b$ and $c$; CAO
---3 The discrete random variable $R$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
$\boldsymbol { r }$ & - 2 & 0 & $a$ & 4 \\
\hline
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$ & 0.3 & $b$ & $c$ & 0.1 \\
\hline
\end{tabular}
\end{center}
It is known that $\mathrm { E } ( R ) = 0.2$ and $\operatorname { Var } ( R ) = 3.56$\\
Find the values of $a , b$ and $c$.\\[0pt]
[4 marks]\\
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics Q3 [4]}}