| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| 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 | Moderate -0.3 This is a straightforward continuous probability distribution question requiring standard techniques: integrating the pdf to find the constant (using ∫f(x)dx=1), calculating E(X) by integration, finding the median from P(X<m)=0.5, and computing a probability. All steps are routine applications of S2 formulas with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{2} \times a \times \frac{a}{2} = 1\) or \(\frac{1}{2}\int_0^a x\,dx = 1\) | M1 | Attempt at triangle area or integral \(f(x)\) and \(= 1\) |
| \(\frac{a^2}{4} = 1\) OE | ||
| \(a = 2\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{2}\int_0^2 x^2\,dx\) | M1 | Attempt integral \(xf(x)\) |
| \(= \left[\frac{x^3}{6}\right]_0^2\) | M1 | Correct integral and limits 0 to their \(a\) |
| \(= \frac{8}{6} = \frac{4}{3}\) | A1 | AG, CWO |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P\!\left(X < \frac{4}{3}\right) = \frac{1}{2}\int_0^{4/3} x\,dx\) | M1 | Attempt integral \(f(x)\) between correct limits |
| \(= \frac{4}{9}\) | A1 | or \(\frac{5}{9}\) |
| \(P(E(X) < X < m) = \frac{1}{2} - \frac{4}{9}\) | M1 | or \(\frac{5}{9} - \frac{1}{2}\) |
| \(\frac{1}{18}\) | A1 | |
| Alternative method: | ||
| Attempt to find \(m\) | M1 | |
| \(m = \sqrt{2}\) | A1 | |
| Integrate \(f(x)\) between \(\frac{4}{3}\) and \(\sqrt{2}\) | M1 | |
| \(\frac{1}{18}\) | A1 | |
| 4 |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{2} \times a \times \frac{a}{2} = 1$ **or** $\frac{1}{2}\int_0^a x\,dx = 1$ | M1 | Attempt at triangle area or integral $f(x)$ **and** $= 1$ |
| $\frac{a^2}{4} = 1$ OE | | |
| $a = 2$ | A1 | |
| | **2** | |
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{2}\int_0^2 x^2\,dx$ | M1 | Attempt integral $xf(x)$ |
| $= \left[\frac{x^3}{6}\right]_0^2$ | M1 | Correct integral and limits 0 to their $a$ |
| $= \frac{8}{6} = \frac{4}{3}$ | A1 | AG, CWO |
| | **3** | |
## Question 4(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P\!\left(X < \frac{4}{3}\right) = \frac{1}{2}\int_0^{4/3} x\,dx$ | M1 | Attempt integral $f(x)$ between correct limits |
| $= \frac{4}{9}$ | A1 | or $\frac{5}{9}$ |
| $P(E(X) < X < m) = \frac{1}{2} - \frac{4}{9}$ | M1 | or $\frac{5}{9} - \frac{1}{2}$ |
| $\frac{1}{18}$ | A1 | |
| **Alternative method:** | | |
| Attempt to find $m$ | M1 | |
| $m = \sqrt{2}$ | A1 | |
| Integrate $f(x)$ between $\frac{4}{3}$ and $\sqrt{2}$ | M1 | |
| $\frac{1}{18}$ | A1 | |
| | **4** | |4 A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { 1 } { 2 } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a constant.\\
(i) Find $a$.\\
(ii) Show that $\mathrm { E } ( X ) = \frac { 4 } { 3 }$.\\
The median of $X$ is denoted by $m$.\\
(iii) Find $\mathrm { P } ( \mathrm { E } ( X ) < X < m )$.\\
\hfill \mbox{\textit{CAIE S2 2019 Q4 [9]}}