| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | PDF to CDF derivation |
| Difficulty | Standard +0.3 This is a standard S2 question requiring routine integration of a piecewise PDF to find the CDF, plus straightforward conditional probability. While multi-part with 6 sections, each step follows textbook procedures: sketching a given function, identifying the mode by inspection, verifying a probability by integration, deriving F(x) by integrating each piece, applying conditional probability formula, and solving a quadratic. No novel insight required, just careful execution of standard techniques. |
| Spec | 2.03d Calculate conditional probability: from first principles5.03a 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 |
7. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by
$$f ( x ) = \begin{cases} \frac { 1 } { 20 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 10 } ( 6 - x ) & 2 < x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $\mathrm { f } ( x )$ for all values of $x$.
\item Write down the mode of $X$.
\item Show that $\mathrm { P } ( X > 2 ) = 0.8$
\item Define fully the cumulative distribution function $\mathrm { F } ( x )$.
Given that $\mathrm { P } ( X < a \mid X > 2 ) = \frac { 5 } { 8 }$
\item find the value of $\mathrm { F } ( a )$.
\item Hence, or otherwise, find the value of $a$. Give your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2017 Q7 [14]}}