| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Explain why not valid PDF |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing basic PDF properties: ordering medians by visual inspection of symmetry/skewness, verifying E(X) through standard integration, and calculating probabilities using the CDF. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(m_X,\ m_Y,\ m_Z,\ m_W\) or \(X, Y, Z, W\) | B2 [2] | B1 if two adjacent means interchanged, i.e. \(m_Y, m_X, m_Z, m_W\) or \(m_X, m_Z, m_Y, m_W\) or \(m_X, m_Y, m_W, m_Z\). B1 for correct order reversed. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_0^3 \frac{4}{81} x^4\, dx\) | M1 | Attempt int \(xf(x)\). Ignore limits |
| \(= \left[\frac{4}{81} \cdot \frac{x^5}{5}\right]_0^3\) | A1 | Correct integration and limits (condone missing 4/81) |
| \(= \frac{4}{81} \times \frac{3^3}{5}\) or \(\frac{4}{81} \times \frac{243}{5}\) or \(\frac{972}{405}\) oe | Must see correct expression as well as \(\frac{12}{5}\) or 2.4 | |
| \(= \frac{12}{5}\) or \(2.4\) AG | A1 [3] | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_{2.4}^3 \frac{4}{81} x^3\, dx\) \(\quad\) or \(\quad\) \(1 - \int_0^{2.4} \frac{4}{81} x^3\, dx\) | M1 | Attempt int \(f(x)\) ignore limits |
| \(= \left[\frac{4}{81} \cdot \frac{x^4}{4}\right]_{2.4}^3\) \(\quad\) or \(\quad\) \(1 - \left[\frac{4}{81} \cdot \frac{x^4}{4}\right]_0^{2.4}\) | A1 | Correct integration and limits (condone missing 4/81) |
| \(= 1 - \frac{4}{81} \times \frac{2.4^4}{4}\) oe | ||
| \(= \frac{369}{625}\) or \(0.59(0)\) (3 sf) | A1 [3] | As final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| 1 | B1 [1] |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $m_X,\ m_Y,\ m_Z,\ m_W$ or $X, Y, Z, W$ | B2 [2] | B1 if two adjacent means interchanged, i.e. $m_Y, m_X, m_Z, m_W$ or $m_X, m_Z, m_Y, m_W$ or $m_X, m_Y, m_W, m_Z$. B1 for correct order reversed. |
### Part (ii)(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_0^3 \frac{4}{81} x^4\, dx$ | M1 | Attempt int $xf(x)$. Ignore limits |
| $= \left[\frac{4}{81} \cdot \frac{x^5}{5}\right]_0^3$ | A1 | Correct integration and limits (condone missing 4/81) |
| $= \frac{4}{81} \times \frac{3^3}{5}$ or $\frac{4}{81} \times \frac{243}{5}$ or $\frac{972}{405}$ oe | | Must see correct expression as well as $\frac{12}{5}$ or 2.4 |
| $= \frac{12}{5}$ or $2.4$ AG | A1 [3] | No errors seen |
### Part (ii)(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_{2.4}^3 \frac{4}{81} x^3\, dx$ $\quad$ or $\quad$ $1 - \int_0^{2.4} \frac{4}{81} x^3\, dx$ | M1 | Attempt int $f(x)$ ignore limits |
| $= \left[\frac{4}{81} \cdot \frac{x^4}{4}\right]_{2.4}^3$ $\quad$ or $\quad$ $1 - \left[\frac{4}{81} \cdot \frac{x^4}{4}\right]_0^{2.4}$ | A1 | Correct integration and limits (condone missing 4/81) |
| $= 1 - \frac{4}{81} \times \frac{2.4^4}{4}$ oe | | |
| $= \frac{369}{625}$ or $0.59(0)$ (3 sf) | A1 [3] | As final answer |
### Part (ii)(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| 1 | B1 [1] | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534}\\
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123}\\
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534}\\
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128}
The diagrams show the probability density functions of four random variables $W , X , Y$ and $Z$. Each of the four variables takes values between 0 and 3 only, and their medians are $m _ { W } , m _ { X } , m _ { Y }$ and $m _ { Z }$ respectively.\\
(i) List $m _ { W } , m _ { X } , m _ { Y }$ and $m _ { Z }$ in order of size, starting with the largest.\\
(ii) The probability density function of $X$ is given by
$$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { E } ( X ) = \frac { 12 } { 5 }$.
\item Calculate $\mathrm { P } ( X > \mathrm { E } ( X ) )$.
\item Write down the value of $\mathrm { P } ( X < 2 \mathrm { E } ( X ) )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2016 Q6 [9]}}