| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 15 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Conditional probability with PDF |
| Difficulty | Standard +0.8 This is a multi-part Further Maths statistics question requiring integration of a power function PDF, finding k using normalization, deriving a CDF, applying conditional probability, and analyzing variance existence conditions. While the techniques are standard (integration, conditional probability formula), the question requires careful algebraic manipulation and understanding of when moments exist for power-law distributions, making it moderately challenging but within typical Further Stats scope. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
4 The continuous random variable $X$ has probability density function
$$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
where $n$ and $k$ are constants and $n$ is an integer greater than 1 .
\begin{enumerate}[label=(\alph*)]
\item Find $k$ in terms of $n$.
\item \begin{enumerate}[label=(\roman*)]
\item When $n = 4$, find the cumulative distribution function of $X$.
\item Hence determine $\mathrm { P } ( X > 7 \mid X > 5 )$ when $n = 4$.
\end{enumerate}\item Determine the values of $n$ for which $\operatorname { Var } ( X )$ is not defined.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2021 Q4 [15]}}