| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find or specify CDF |
| Difficulty | Standard +0.3 This is a standard S2 question requiring routine integration to find a CDF and application of conditional probability. Part (a) is straightforward sketching, part (b) involves integrating two piecewise functions (one polynomial, one linear) with clear boundaries, and part (c) applies the standard conditional probability formula F(4)-F(3)/F(4)-F(2). All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| 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 percentiles |
7 The continuous random variable $X$ has the probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\
\frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5 \\
0 & \text { otherwise }
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of f.
\item Prove that the cumulative distribution function of $X$ for $2 \leqslant x \leqslant 5$ can be written in the form
$$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
\item Hence, or otherwise, determine $\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2009 Q7 [12]}}