| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Expectation of reciprocals and nonlinear functions |
| Difficulty | Standard +0.8 This S2 question requires multiple non-routine calculations: verifying E(1/X) involves summing reciprocals weighted by probabilities, finding Var(1/X) requires E(1/X²), and part (c) demands finding E and Var of the product (X+3)(1/X) = 1 + 3/X, requiring careful algebraic manipulation and application of variance properties. While the individual techniques are A-level standard, the combination of reciprocal transformations and the multi-step product calculation elevates this above typical textbook exercises. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.03d E(g(X)): general expectation formula |
5 A discrete random variable $X$ has the probability distribution
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l }
\frac { x } { 20 } & x = 1,2,3,4,5 \\
\frac { x } { 24 } & x = 6 \\
0 & \text { otherwise }
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Calculate $\mathrm { P } ( X \geqslant 5 )$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }$.
\item Hence, or otherwise, show that $\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036$, correct to three decimal places.
\end{enumerate}\item Calculate the mean and the variance of $A$, the area of rectangles having sides of length $X + 3$ and $\frac { 1 } { X }$.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2008 Q5 [12]}}