| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find expectation E(X) |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard pdf techniques. Part (i) requires routine integration of x·f(x) and solving F(m)=0.5 for the median—both mechanical calculations with the given pdf. Part (ii) involves computing E(Y²) which diverges, a slightly less routine observation but still a direct application of the variance formula. All parts follow standard textbook methods with no novel problem-solving required, making this slightly easier than average. |
| Spec | 5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
\begin{enumerate}[label=(\roman*)]
\item The continuous random variable $X$ has the probability density function
$$f(x) = \begin{cases}
\frac{1}{2\sqrt{x}} & 1 < x < 4, \\
0 & \text{otherwise}.
\end{cases}$$
Find \begin{enumerate}[label=(\alph*)]
\item E($X$), [3]
\item the median of $X$. [3]
\end{enumerate}
\item The continuous random variable $Y$ has the probability density function
$$g(y) = \begin{cases}
\frac{1.5}{y^{2.5}} & y > 1, \\
0 & \text{otherwise}.
\end{cases}$$
Given that E($Y$) = 3, show that Var($Y$) is not finite. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2012 Q7 [9]}}