| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Sketch or interpret PDF graph |
| Difficulty | Moderate -0.3 This question tests interpretation of PDF graphs and basic probability concepts (median, standard deviation, calculating probabilities from PDFs). While it requires understanding of continuous distributions, the tasks are largely visual interpretation and one straightforward integration. The conceptual demands are moderate for S2 level, making it slightly easier than average. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X\) or 5 | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(V\) or 3 | B1 | Should mention values or prob; not just graph or spread e.g. not "More spread" |
| Higher and lower values more likely or there are more higher and lower values or more prob at both extremes | B1dep [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{2+1}{2} \times 0.5\) or \(\int_0^{0.5}(2-2x)\,dx\) | M1 | ('or' method requires linear function and correct limits) |
| \(= 0.75\) | A1 [2] | CWO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_0^1 ax^n\,dx = 1\) | M1 | Attempt integ of correct form \(= 1\) (ignore limits) |
| \(\left[\dfrac{ax^{n+1}}{n+1}\right]_0^1 = 1\) | A1 | Correct integrand & limits |
| \(\dfrac{a}{n+1} = 1\) | A1 | No errors seen |
| \((a = n+1\) AG)$ | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_0^1 ax^{n+1}\,dx = \dfrac{5}{6}\) oe | M1* | Integral of form \(\int xf(x)\,dx = \dfrac{5}{6}\), ignore limits |
| \(\left[\dfrac{ax^{n+2}}{n+2}\right]_0^1 = \dfrac{5}{6}\) oe | A1 | Correct integrand & limits |
| \(\dfrac{a}{n+2} = \dfrac{5}{6}\); \((6a = 5n + 10)\) | M1dep | Attempt to use \(a = n+1\) within 2nd equation to get an equation in \(n\) (or \(a\)) |
| \(a = 5,\ n = 4\) | A1 [4] |
## Question 7:
### Part (i)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X$ or 5 | B1 [1] | |
### Part (i)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $V$ or 3 | B1 | Should mention values or prob; not just graph or spread e.g. not "More spread" |
| Higher and lower values more likely or there are more higher and lower values or more prob at both extremes | B1dep [2] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{2+1}{2} \times 0.5$ or $\int_0^{0.5}(2-2x)\,dx$ | M1 | ('or' method requires linear function and correct limits) |
| $= 0.75$ | A1 [2] | CWO |
### Part (iii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_0^1 ax^n\,dx = 1$ | M1 | Attempt integ of correct form $= 1$ (ignore limits) |
| $\left[\dfrac{ax^{n+1}}{n+1}\right]_0^1 = 1$ | A1 | Correct integrand & limits |
| $\dfrac{a}{n+1} = 1$ | A1 | No errors seen |
| $(a = n+1$ **AG**)$ | [3] | |
### Part (iii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_0^1 ax^{n+1}\,dx = \dfrac{5}{6}$ oe | M1* | Integral of form $\int xf(x)\,dx = \dfrac{5}{6}$, ignore limits |
| $\left[\dfrac{ax^{n+2}}{n+2}\right]_0^1 = \dfrac{5}{6}$ oe | A1 | Correct integrand & limits |
| $\dfrac{a}{n+2} = \dfrac{5}{6}$; $(6a = 5n + 10)$ | M1dep | Attempt to use $a = n+1$ within 2nd equation to get an equation in $n$ (or $a$) |
| $a = 5,\ n = 4$ | A1 [4] | |7
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_385_385_982_246}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_385_380_982_669}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_390_378_977_1087}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_390_391_977_1503}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_392_391_1475_370}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_392_387_1475_872}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3629cb5c-fb4c-43b1-b43f-ce321411b814-3_392_389_1475_1375}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
Each of the random variables $T , U , V , W , X , Y$ and $Z$ takes values between 0 and 1 only. Their probability density functions are shown in Figs 1 to 7 respectively.\\
(i) (a) Which of these variables has the largest median?\\
(b) Which of these variables has the largest standard deviation? Explain your answer.\\
(ii) Use Fig. 2 to find $\mathrm { P } ( U < 0.5 )$.\\
(iii) The probability density function of $X$ is given by
$$f ( x ) = \begin{cases} a x ^ { n } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $n$ are positive constants.\\
(a) Show that $a = n + 1$.\\
(b) Given that $\mathrm { E } ( X ) = \frac { 5 } { 6 }$, find $a$ and $n$.
\hfill \mbox{\textit{CAIE S2 2011 Q7 [12]}}