| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Topic | Continuous Probability Distributions and Random Variables |
| Difficulty | Challenging +1.2 This is a standard Further Maths probability question requiring integration of a given pdf. Parts (i)-(iii) involve routine calculations: verifying the median through integration, finding E(X) using standard expectation formula, and computing variance. Part (iv) adds conditional probability but with a helpful hint about tan(π/8). The pdf involves arctan integration which is A-level standard, and all techniques are well-practiced in Further Maths courses. The multi-part structure and 13 marks indicate moderate length, but no novel insight is required—just careful application of standard methods. |
| Spec | 2.03c Conditional probability: using diagrams/tables5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
The continuous random variable $X$ has probability density function given by
$$f(x) = \begin{cases} \frac{4}{\pi(1+x^2)} & 0 \leq x \leq 1, \\ 0 & \text{otherwise.} \end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item Verify that the median value of $X$ lies between 0.41 and 0.42. [3]
\item Show that E$(X) = \frac{2}{\pi}\ln 2$. [2]
\item Find Var$(X)$. [5]
\item Given that $\tan\frac{1}{8}\pi = \sqrt{2} - 1$, find the exact value of P($X > \frac{1}{4}\sqrt{3}|X > \sqrt{2} - 1$). [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2014 Q6 [13]}}