| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| 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 Part (a) requires recognizing PDF symmetry (a conceptual insight but straightforward once seen). Parts (b) and (c) involve standard integration techniques for finding k and variance. Part (d) requires understanding order statistics and calculating P(max > 2.5) = 1 - P(all ≤ 2.5)³, which is a moderately challenging application but follows standard probability methods. Overall slightly easier than average due to the symmetry shortcut and routine calculus. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Quadratic curve, hence symmetrical | B1 | OE. Allow sketch and 'symmetrical' or just 'curve symmetrical' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(-k\int_1^3 (x^2 - 4x + 3)\,dx = 1\) | M1 | Attempt to integrate \(f(x)\) and \('= 1'\). Ignore limits at this stage |
| \(-k\left[\frac{x^3}{3} - 2x^2 + 3x\right]_1^3\) | A1 | Fully correct expression (correct integration and limits) |
| \(-k \times \left[0 - \frac{4}{3}\right] = 1\) or \(k \times \frac{4}{3} = 1\), \(\left[k = \frac{3}{4}\right]\) | A1 | AG, OE. Correctly substitute limits and \('= 1'\) and correctly obtain result with no errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(-\frac{3}{4}\int_1^3 (x^4 - 4x^3 + 3x^2)\,dx\) | M1 | Attempt to integrate \(x^2 f(x)\) from 1 to 3 |
| \(-\frac{3}{4} \times \left[\frac{x^5}{5} - x^4 + x^3\right]_1^3\) | A1 | Correct integration and limits |
| \(\left[= \frac{3}{4} \times \frac{28}{5} = \frac{21}{5}\right]\) | ||
| \(\left[\frac{21}{5} - 2^2\right] = 0.2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(-\frac{3}{4}\int_{2.5}^3 (x^2 - 4x + 3)\,dx\) | M1 | OE. Attempt to integrate \(f(x)\), from 2.5 to 3 (or 1 to 2.5) |
| \(= -\frac{3}{4} \times \left[\frac{x^3}{3} - 2x^2 + 3x\right]_{2.5}^3 = \frac{5}{32}\) or \(0.15625\) | A1 | |
| \(1 - \left(1 - '\frac{5}{32}'\right)^3\) | M1 | OE. FT *their* \(\frac{5}{32}\) |
| \(= 0.399\) (3 sf) | A1 |
## Question 6:
### Part 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Quadratic curve, hence symmetrical | B1 | OE. Allow sketch and 'symmetrical' or just 'curve symmetrical' |
### Part 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $-k\int_1^3 (x^2 - 4x + 3)\,dx = 1$ | M1 | Attempt to integrate $f(x)$ and $'= 1'$. Ignore limits at this stage |
| $-k\left[\frac{x^3}{3} - 2x^2 + 3x\right]_1^3$ | A1 | Fully correct expression (correct integration and limits) |
| $-k \times \left[0 - \frac{4}{3}\right] = 1$ or $k \times \frac{4}{3} = 1$, $\left[k = \frac{3}{4}\right]$ | A1 | AG, OE. Correctly substitute limits and $'= 1'$ and correctly obtain result with no errors seen |
### Part 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $-\frac{3}{4}\int_1^3 (x^4 - 4x^3 + 3x^2)\,dx$ | M1 | Attempt to integrate $x^2 f(x)$ from 1 to 3 |
| $-\frac{3}{4} \times \left[\frac{x^5}{5} - x^4 + x^3\right]_1^3$ | A1 | Correct integration and limits |
| $\left[= \frac{3}{4} \times \frac{28}{5} = \frac{21}{5}\right]$ | | |
| $\left[\frac{21}{5} - 2^2\right] = 0.2$ | A1 | |
### Part 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $-\frac{3}{4}\int_{2.5}^3 (x^2 - 4x + 3)\,dx$ | M1 | OE. Attempt to integrate $f(x)$, from 2.5 to 3 (or 1 to 2.5) |
| $= -\frac{3}{4} \times \left[\frac{x^3}{3} - 2x^2 + 3x\right]_{2.5}^3 = \frac{5}{32}$ or $0.15625$ | A1 | |
| $1 - \left(1 - '\frac{5}{32}'\right)^3$ | M1 | OE. FT *their* $\frac{5}{32}$ |
| $= 0.399$ (3 sf) | A1 | |
---6 In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable $X$ with probability density function
$$f ( x ) = \begin{cases} - k ( x - 1 ) ( x - 3 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Without calculation, explain why $\mathrm { E } ( X ) = 2$.
\item Show that $k = \frac { 3 } { 4 }$.
\item Find $\operatorname { Var } ( X )$.\\
One turn consists of rolling the ball 3 times and noting the largest value of $X$ obtained. If this largest value is greater than 2.5, the player scores a point.
\item Find the probability that on a particular turn the player scores a point.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q6 [11]}}