| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Find quantiles from CDF |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard CDF operations: finding median by solving F(x)=0.5, differentiating to get PDF, evaluating probabilities, and applying independence for multiple batteries. All techniques are routine with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
4. The lifetime, $X$, in tens of hours, of a battery has a cumulative distribution function $\mathrm { F } ( x )$ given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < 1 \\
\frac { 4 } { 9 } \left( x ^ { 2 } + 2 x - 3 \right) & 1 \leqslant x \leqslant 1.5 \\
1 & x > 1.5
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Find the median of $X$, giving your answer to 3 significant figures.
\item Find, in full, the probability density function of the random variable $X$.
\item Find $\mathrm { P } ( X \geqslant 1.2 )$
A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
\item Find the probability that the lantern will still be working after 12 hours.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [10]}}