| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Geometric/graphical PDF with k |
| Difficulty | Standard +0.8 This Further Pure question requires finding k from the normalization condition, deriving the CDF by integrating piecewise linear functions, then applying transformation of variables (Y=X²) requiring the Jacobian method and careful domain mapping. The median calculation involves solving a cubic equation. Multiple sophisticated techniques beyond standard A-level, but systematic application of learned methods. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles5.03g Cdf of transformed variables |
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{020ebd88-b920-40ce-84cf-5c26d45e2935-5_383_839_1635_651}
\end{center}
The continuous random variable $X$ takes values in the interval $0 \leqslant x \leqslant 3$ only. For $0 \leqslant x \leqslant 3$ the graph of its probability density function f consists of two straight line segments meeting at the point $( 1 , k )$, as shown in the diagram. Find $k$ and hence show that the distribution function F is given by
$$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$
The random variable $Y$ is given by $Y = X ^ { 2 }$. Find\\
(i) the probability density function of $Y$,\\
(ii) the median value of $Y$.
\hfill \mbox{\textit{CAIE FP2 2011 Q11 OR}}