| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from table |
| Difficulty | Moderate -0.8 This is a straightforward S2 question requiring standard application of variance formula from a probability distribution table, followed by routine use of E(aX+b) and Var(aX+b) properties. All steps are mechanical with no problem-solving or insight required—easier than average A-level. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(\boldsymbol { x }\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\) | 0.1 | 0.35 | 0.25 | 0.2 | 0.1 |
5 Aiden takes his car to a garage for its MOT test. The probability that his car will need to have $X$ tyres replaced is shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$ & 0.1 & 0.35 & 0.25 & 0.2 & 0.1 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Show that the mean of $X$ is 1.85 and calculate the variance of $X$.
\item The charge for the MOT test is $\pounds c$ and the cost of each new tyre is $\pounds n$. The total amount that Aiden must pay the garage is $\pounds T$.
\begin{enumerate}[label=(\roman*)]
\item Express $T$ in terms of $c , n$ and $X$.
\item Hence, using your results from part (a), find expressions for $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2013 Q5 [9]}}