| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Compare uniform with other distributions |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring a sketch of two pdfs (one uniform, one symmetric polynomial), a verbal comparison of distributions (testing understanding that T is more concentrated at extremes), and a routine variance calculation using integration. All techniques are standard for this module with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
5 The continuous random variables $S$ and $T$ have probability density functions as follows.
$$\begin{array} { l l }
S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2 \\
0 & \text { otherwise } \end{cases} \\
T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2 \\
0 & \text { otherwise } \end{cases}
\end{array}$$
(i) Sketch, on the same axes, the graphs of f and g .\\
(ii) Describe in everyday terms the difference between the distributions of the random variables $S$ and $T$. (Answers that comment only on the shapes of the graphs will receive no credit.)\\
(iii) Calculate the variance of $T$.
\hfill \mbox{\textit{OCR S2 2009 Q5 [9]}}