| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2021 |
| Session | January |
| Marks | 7 |
| Topic | Cumulative distribution functions |
| Type | Mode from PDF |
| Difficulty | Standard +0.3 This is a straightforward Further Maths statistics question requiring standard techniques: finding k using F(2)=1, differentiating F to get f, then computing E(X) by integration and finding the mode by maximizing f. All steps are routine applications of well-practiced methods with no conceptual surprises, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
The continuous random variable $X$ has cumulative distribution function given by
$$F(x) = \begin{cases}
0 & x \leq 0 \\
k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\
1 & x > 2
\end{cases}$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac{1}{2}$ [1]
\item Showing your working clearly, use calculus to find
\begin{enumerate}[label=(\roman*)]
\item E($X$)
\item the mode of $X$
\end{enumerate}
[6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2021 Q4 [7]}}