| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Symmetry property of PDF |
| Difficulty | Standard +0.3 This question tests standard PDF properties including symmetry recognition for E(X), variance calculation via integration, and using symmetry for probability calculations. Part (b) involves routine median and normalization conditions. While requiring multiple techniques, all are textbook applications with no novel problem-solving—slightly easier than average due to the symmetry shortcut provided by the diagram. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03d E(g(X)): general expectation formula5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(X) = 1.5\) | B1 | |
| \(\frac{2}{9}\int_0^3 (3x^3 - x^4)\,dx\) | M1 | Attempt \(\int x^2 f(x)\), ignore limits |
| \(= \frac{2}{9}\left[\frac{3x^4}{4} - \frac{x^5}{5}\right]_0^3\) | ||
| \(= \frac{2}{9}\left[\frac{243}{4} - \frac{243}{5}\right]\) \((= 2.7)\) | M1 | Sub correct limits into correct integral |
| \(\text{Var}(X) = 2.7 - 1.5^2 = 0.45\) oe | A1\(\checkmark\) [4] | Ft their \(E(X)\), but no ft for negative Var |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.5\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(1 - \frac{13}{27}\right) \div 2\) | M1 | or \(\frac{2}{9}\int_2^3 (3x - x^2)\,dx\) oe |
| \(= \frac{7}{27}\) or \(0.259\) | A1 [2] | As final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2} \times 2 \times 2a = \frac{1}{2}\) or \(\int_0^2 ax\,dx = \frac{1}{2}\) | M1 | Attempt correct equation in \(a\) |
| \(a = \frac{1}{4}\) | A1 | |
| \(\frac{1}{2} \times b \times \frac{1}{4}b = 1\) or \(\int_0^b \frac{1}{4}x\,dx = 1\) or \(b = 2\times\sqrt{2}\) | M1 | or \(\frac{1}{2} \times b \times ab = 1\) or \(\int_0^b ax\,dx = 1\); attempt correct equation in \(a\) and \(b\) |
| \(b = 2\sqrt{2}\) | A1\(\checkmark\) [4] | Allow \(b = \sqrt{8}\) or 2.83 (3 sf); ft incorrect \(a\), both Ms needed |
## Question 7:
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(X) = 1.5$ | **B1** | |
| $\frac{2}{9}\int_0^3 (3x^3 - x^4)\,dx$ | **M1** | Attempt $\int x^2 f(x)$, ignore limits |
| $= \frac{2}{9}\left[\frac{3x^4}{4} - \frac{x^5}{5}\right]_0^3$ | | |
| $= \frac{2}{9}\left[\frac{243}{4} - \frac{243}{5}\right]$ $(= 2.7)$ | **M1** | Sub correct limits into correct integral |
| $\text{Var}(X) = 2.7 - 1.5^2 = 0.45$ oe | **A1$\checkmark$** [4] | Ft their $E(X)$, but no ft for negative Var |
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.5$ | **B1** [1] | |
### Part (a)(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(1 - \frac{13}{27}\right) \div 2$ | **M1** | or $\frac{2}{9}\int_2^3 (3x - x^2)\,dx$ oe |
| $= \frac{7}{27}$ or $0.259$ | **A1** [2] | As final answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2} \times 2 \times 2a = \frac{1}{2}$ or $\int_0^2 ax\,dx = \frac{1}{2}$ | **M1** | Attempt correct equation in $a$ |
| $a = \frac{1}{4}$ | **A1** | |
| $\frac{1}{2} \times b \times \frac{1}{4}b = 1$ or $\int_0^b \frac{1}{4}x\,dx = 1$ or $b = 2\times\sqrt{2}$ | **M1** | or $\frac{1}{2} \times b \times ab = 1$ or $\int_0^b ax\,dx = 1$; attempt correct equation in $a$ and $b$ |
| $b = 2\sqrt{2}$ | **A1$\checkmark$** [4] | Allow $b = \sqrt{8}$ or 2.83 (3 sf); ft incorrect $a$, both Ms needed |
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**Total for paper: 50**7
\begin{enumerate}[label=(\alph*)]
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{3f1a0c67-03a4-4b4f-99c0-4336ba7d56b0-3_255_643_264_790}
The diagram shows the graph of the probability density function, f , of a random variable $X$, where
$$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item State the value of $\mathrm { E } ( X )$ and find $\operatorname { Var } ( X )$.
\item State the value of $\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )$.
\item Given that $\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }$, find $\mathrm { P } ( X > 2 )$.
\end{enumerate}\item A random variable, $W$, has probability density function given by
$$\mathrm { g } ( w ) = \begin{cases} a w & 0 \leqslant w \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $b$ are constants. Given that the median of $W$ is 2 , find $a$ and $b$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2016 Q7 [11]}}