| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Probability distribution from formula |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution question requiring only standard calculations: summing probabilities to find k, computing E(X) and Var(X) using formulas, and applying linear transformations. All steps are routine S2 techniques with no problem-solving or novel insight required, making it easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable $X$.
The probability distribution of $X$ is defined by
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l }
\frac { 1 } { x + 2 } & x = 1,2,3,4 \\
k & x = 5 \\
0 & \text { otherwise }
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $k = \frac { 1 } { 20 }$.
\item Calculate the value of $\mathrm { E } ( X )$.
\item Show that $\operatorname { Var } ( X ) = 1.5275$.
\item Find the probability that Isaac collects more than 2 treasures.
\end{enumerate}\item The number of points that Isaac scores for collecting treasures is $Y$ where
$$Y = 100 X - 50$$
Calculate the mean and the standard deviation of $Y$.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2013 Q5 [13]}}