| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | March |
| Marks | 10 |
| 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 symmetry properties of PDFs and routine integration. Part (a) requires understanding that symmetry about x=2 means P(X>5) = P(X<-1), leading to a straightforward calculation. Part (b) involves integrating a quadratic to find k (standard normalization). Part (c) requires computing variance using E(X²) - [E(X)]², which is routine integration of polynomials. All techniques are textbook exercises with no novel insight required, making this easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2} - \frac{117}{256} \quad \left[= \frac{11}{256}\right]\) | M1 | For use of symmetry about \(x = 2\), oe. E.g. \(\left(1 - 2\times\frac{117}{256}\right)\div 2\) or \(\frac{1}{2} + \frac{117}{256}\) |
| \(1 - \frac{11}{256}\) or \(2\times\frac{117}{256} + \frac{11}{256} = \frac{245}{256}\) AG | A1 | Any correct numerical expression seen leading to AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(k\int_{2}^{5}(12+4x-x^2)dx\) | M1 | Attempt to integrate \(f(x)\) with any limits. |
| \(= k\left[12x+2x^2-\frac{x^3}{3}\right]_{2}^{5}\) | M1 | Use of limits 2 and 5 and equating *their* integration attempt to \(\frac{117}{256}\). Or limits \(-2\) and 6 equated to 1. Or limits \(-1\) and 6 equated to \(\frac{245}{256}\). Or limits \(-1\) to 5 equated to \(\frac{234}{256}\). Oe. No mixed methods. |
| \(39k = \frac{117}{256}\) | M1 | |
| \(k = \frac{3}{256}\) or \(0.0117\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([2+x-x^2=0]\), \(x=-1\) and \(x=2\) seen or implied. Domain is \(-1 \leqslant x \leqslant 2\) | B1 | |
| Mean \(= 0.5\) | B1 | |
| \(\frac{2}{9}\int_{-1}^{2}(2x^2+x^3-x^4)dx\) | *M1 | Attempt to integrate \(x^2 g(x)\) with any limits. |
| \(= \frac{2}{9}\left[\frac{2}{3}x^3+\frac{x^4}{4}-\frac{x^5}{5}\right]_{-1}^{2}\) | ||
| \([= 0.7]\) | ||
| *their* \('0.7'\) \(-\) *their* \('0.5'^2\) | DM1 | Subtract *their* mean\(^2\) from *their* \(\int x^2 g(x)dx\) (both must be numerical). |
| \(= 0.45\) or \(\frac{9}{20}\) | A1 | |
| Total: 5 |
## Question 6(a):
| $\frac{1}{2} - \frac{117}{256} \quad \left[= \frac{11}{256}\right]$ | M1 | For use of symmetry about $x = 2$, oe. E.g. $\left(1 - 2\times\frac{117}{256}\right)\div 2$ or $\frac{1}{2} + \frac{117}{256}$ |
|---|---|---|
| $1 - \frac{11}{256}$ or $2\times\frac{117}{256} + \frac{11}{256} = \frac{245}{256}$ **AG** | A1 | Any correct numerical expression seen leading to AG |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $k\int_{2}^{5}(12+4x-x^2)dx$ | M1 | Attempt to integrate $f(x)$ with any limits. |
| $= k\left[12x+2x^2-\frac{x^3}{3}\right]_{2}^{5}$ | M1 | Use of limits 2 and 5 and equating *their* integration attempt to $\frac{117}{256}$. Or limits $-2$ and 6 equated to 1. Or limits $-1$ and 6 equated to $\frac{245}{256}$. Or limits $-1$ to 5 equated to $\frac{234}{256}$. Oe. No mixed methods. |
| $39k = \frac{117}{256}$ | M1 | |
| $k = \frac{3}{256}$ or $0.0117$ | A1 | |
| **Total: 3** | | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[2+x-x^2=0]$, $x=-1$ and $x=2$ seen or implied. Domain is $-1 \leqslant x \leqslant 2$ | B1 | |
| Mean $= 0.5$ | B1 | |
| $\frac{2}{9}\int_{-1}^{2}(2x^2+x^3-x^4)dx$ | *M1 | Attempt to integrate $x^2 g(x)$ with any limits. |
| $= \frac{2}{9}\left[\frac{2}{3}x^3+\frac{x^4}{4}-\frac{x^5}{5}\right]_{-1}^{2}$ | | |
| $[= 0.7]$ | | |
| *their* $'0.7'$ $-$ *their* $'0.5'^2$ | DM1 | Subtract *their* mean$^2$ from *their* $\int x^2 g(x)dx$ (both must be numerical). |
| $= 0.45$ or $\frac{9}{20}$ | A1 | |
| **Total: 5** | | |
---6 The graph of the probability density function f of a random variable $X$ is symmetrical about the line $x = 2$. It is given that $\mathrm { P } ( 2 < X < 5 ) = \frac { 117 } { 256 }$.
\begin{enumerate}[label=(\alph*)]
\item Using only this information show that $\mathrm { P } ( X > - 1 ) = \frac { 245 } { 256 }$.\\
It is now given that, for $x$ in a suitable domain,
$$f ( x ) = k \left( 12 + 4 x - x ^ { 2 } \right) , \text { where } k \text { is a constant. }$$
\item Find the value of $k$.
\item A different random variable $X$ has probability density function $\mathbf { g } ( x ) = \frac { 2 } { 9 } \left( 2 + x - x ^ { 2 } \right)$. The domain of $X$ is all values of $x$ for which $\mathrm { g } ( x ) \geqslant 0$.
Find $\operatorname { Var } ( X )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{ff3433b0-baab-45e3-845e-56a794739bba-11_63_1547_447_347}
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q6 [10]}}