| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard pdf techniques: part (a)(i) uses integration to find k (routine calculus), part (a)(ii) finds E(X) using the standard formula, and part (b) applies symmetry properties of pdfs with basic probability manipulation. All steps are textbook exercises requiring recall and direct application rather than problem-solving or insight. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k\displaystyle\int_0^{2}(4x - x^2)\,dx = 1\) | M1 | Attempt integral \(f(x)\) and \(= 1\). Ignore limits (must see a power increase for attempted integration) |
| \(k\!\left[\dfrac{4x^2}{2} - \dfrac{x^3}{3}\right]_0^2 = 1\) | A1 | Correct integration and correct limits |
| \(k \times \dfrac{16}{3} = 1\ \left[k = \dfrac{3}{16}\right]\) | A1 | OE AG. Convincingly obtained. At least one interim step. No errors seen |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{3}{16}\int_{0}^{2}(4x^2 - x^3)\,dx\) | M1 | Attempt integral \(xf(x)\). Ignore limits. (must see a power increase for attempted integration). Condone missing \(k\). |
| \(\frac{3}{16}\left[\frac{4x^2}{2} - \frac{x^3}{3}\right]_{0}^{2}\) | A1 | Correct integration and correct limits. Condone missing \(k\). |
| \(\frac{5}{4}\) | A1 | Unsupported correct answer scores SC B2 only. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Symmetrical frequency density graph, 0 to 5, showing area 0.2 to left of \(a\) | B1 | With \(a\) to the left of centre. |
| Either 0.2 between \(5-a\) and 5 or 0.8 between 0 and \(5-a\) | B1 | Shown on graph or stated (\(5-a\) seen or implied). \(a\) must be non-numerical. |
| \([P(2.5 < Y < 5-a)] = 0.3\) | B1 | Must be clearly final answer. \(a\) must be non-numerical. |
| 3 |
## Question 7(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k\displaystyle\int_0^{2}(4x - x^2)\,dx = 1$ | **M1** | Attempt integral $f(x)$ and $= 1$. Ignore limits (must see a power increase for attempted integration) |
| $k\!\left[\dfrac{4x^2}{2} - \dfrac{x^3}{3}\right]_0^2 = 1$ | **A1** | Correct integration and correct limits |
| $k \times \dfrac{16}{3} = 1\ \left[k = \dfrac{3}{16}\right]$ | **A1** | OE AG. Convincingly obtained. At least one interim step. No errors seen |
| | **3** | |
## Question 7(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{3}{16}\int_{0}^{2}(4x^2 - x^3)\,dx$ | **M1** | Attempt integral $xf(x)$. Ignore limits. (must see a power increase for attempted integration). Condone missing $k$. |
| $\frac{3}{16}\left[\frac{4x^2}{2} - \frac{x^3}{3}\right]_{0}^{2}$ | **A1** | Correct integration and correct limits. Condone missing $k$. |
| $\frac{5}{4}$ | **A1** | Unsupported correct answer scores **SC B2** only. |
| | **3** | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Symmetrical frequency density graph, 0 to 5, showing area 0.2 to left of $a$ | **B1** | With $a$ to the left of centre. |
| Either 0.2 between $5-a$ and 5 or 0.8 between 0 and $5-a$ | **B1** | Shown on graph or stated ($5-a$ seen or implied). $a$ must be non-numerical. |
| $[P(2.5 < Y < 5-a)] = 0.3$ | **B1** | Must be clearly final answer. $a$ must be non-numerical. |
| | **3** | |7
\begin{enumerate}[label=(\alph*)]
\item The probability density function of the random variable $X$ is given by
$$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that $k = \frac { 3 } { 16 }$.
\item Find $\mathrm { E } ( X )$.
\end{enumerate}\item The random variable $Y$ has the following properties.
\begin{itemize}
\item $Y$ takes values between 0 and 5 only.
\item The probability density function of $Y$ is symmetrical.
\end{itemize}
Given that $\mathrm { P } ( Y < a ) = 0.2$, find $\mathrm { P } ( 2.5 < Y < 5 - a )$ illustrating your method with a sketch on the axes provided.\\
\includegraphics[max width=\textwidth, alt={}, center]{cea87af9-4b2a-4297-91e9-4eb5744b9e48-11_369_837_621_694}\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q7 [9]}}