| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Geometric/graphical PDF with k |
| Difficulty | Moderate -0.8 This is a straightforward PDF question requiring basic integration. Part (a) uses the fundamental property that total probability equals 1 (likely a simple geometric area calculation from the diagram), and part (b) is a standard E(X) calculation. Both are routine S2 techniques with no problem-solving insight needed. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{2}p(p-1) = 1\) | M1 | For area \(= 1\). For verification methods accept \(\frac{1}{2}\times2\times1=1\) or \(\frac{1}{2}\times2\times(p-1)=1\) or \(\frac{1}{2}\times1\times p=1\) as indication that area\(=1\) |
| \(p = 2\) | A1 | AG - Convincing method and answer. Must see quadratic rearranged to \(=0\) and no errors seen. N.B. Accept convincing verification methods (e.g. statement such as 'assume \(p=2\)' or 'if \(p=2\)' or 'using \(p=2\)' or showing by clear substitution that \(p=2\) fits \(\frac{1}{2}p(p-1)=1\) with clear conclusion) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Gradient \(= 2\), equation of line is \(y = 2x + c\), line passes through \((1,0)\), hence \(c = -2\) | M1 | Award for attempting equation of line \(y=mx+c\) with \(m=2, -2, \frac{1}{2}\) or \(-\frac{1}{2}\) and numerical \(c\) (\(c\neq0\)) |
| \(y = 2x - 2\) | A1 | May be seen in (a). M1 can be implied by correct answer |
| \(2\int_{1}^{2}(x^2 - x)\,dx\) | M1 | For attempting \(\int xf(x)\,dx\). Ignore limits, FT *their* equation |
| \(2\left[\frac{x^3}{3} - \frac{x^2}{2}\right]_1^2\) | A1 FT | Correct integration FT *their* \(f(x)\) and correct limits |
| \(\frac{5}{3}\) or \(1.67\) (3 sf) | A1 |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{2}p(p-1) = 1$ | M1 | For area $= 1$. For verification methods accept $\frac{1}{2}\times2\times1=1$ or $\frac{1}{2}\times2\times(p-1)=1$ or $\frac{1}{2}\times1\times p=1$ as indication that area$=1$ |
| $p = 2$ | A1 | AG - Convincing method and answer. Must see quadratic rearranged to $=0$ and no errors seen. N.B. Accept convincing verification methods (e.g. statement such as 'assume $p=2$' or 'if $p=2$' or 'using $p=2$' or showing by clear substitution that $p=2$ fits $\frac{1}{2}p(p-1)=1$ with clear conclusion) |
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Gradient $= 2$, equation of line is $y = 2x + c$, line passes through $(1,0)$, hence $c = -2$ | M1 | Award for attempting equation of line $y=mx+c$ with $m=2, -2, \frac{1}{2}$ or $-\frac{1}{2}$ and numerical $c$ ($c\neq0$) |
| $y = 2x - 2$ | A1 | May be seen in **(a)**. M1 can be implied by correct answer |
| $2\int_{1}^{2}(x^2 - x)\,dx$ | M1 | For attempting $\int xf(x)\,dx$. Ignore limits, FT *their* equation |
| $2\left[\frac{x^3}{3} - \frac{x^2}{2}\right]_1^2$ | A1 FT | Correct integration FT *their* $f(x)$ and correct limits |
| $\frac{5}{3}$ or $1.67$ (3 sf) | A1 | |3\\
\includegraphics[max width=\textwidth, alt={}, center]{9fca48da-82c3-4ce1-9e0c-93eb9a920f9d-05_456_668_260_735}
The random variable $X$ takes values in the range $1 \leqslant x \leqslant p$, where $p$ is a constant. The graph of the probability density function of $X$ is shown in the diagram.
\begin{enumerate}[label=(\alph*)]
\item Show that $p = 2$.
\item Find $\mathrm { E } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q3 [7]}}