| 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 is a visual interpretation question requiring understanding of PDF properties (median, standard deviation) and basic probability calculations. While it requires conceptual understanding of how PDF shape relates to statistical measures, the actual tasks are straightforward: identifying which distribution has properties based on visual inspection and calculating a simple area under a triangular PDF. This is slightly easier than average as it's primarily conceptual recognition rather than algebraic manipulation. |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (a) \(X\) or 5 | B1 | [1] |
| (b) \(V\) or 3 | B1 | Should mention values or prob |
| Not just graph or spread eg 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] |
| (ii) \(\frac{2 + 1}{2} \times 0.5\) or \(\int_0^{0.5}(2 - 2x)dx\) | M1 | ('or' method requires linear function and correct limits) |
| = 0.75 | A1 | [2] |
| (iii) (a) \(\int ax^n dx = 1\) | M1 | Attempt integ of correct form = 1 (ignore limits) |
| \[\left[\frac{ax^{n+1}}{n+1}\right]_0^1 = 1\] | A1 | Correct integrand & limits |
| \[\frac{a}{n+1} = 1\] | A1 | No errors seen |
| (a = n + 1 AG) | [3] | |
| (b) \(\int ax^n dx = \frac{5}{6}\) oe | M1* | Integral of form \(xf(x)hx = \frac{5}{6}\), ignore limits |
| \[\left[\frac{ax^{n+2}}{n+2}\right]_0^5 \text{ oe}\] | A1 | Correct integrand & limits |
| Answer | Marks | Guidance |
|---|---|---|
| (6a = 5n + 10) | M1dep | Attempt to use \(a = n + 1\) within 2nd equ to get an equ in \(n\) (or \(a\)) |
| a = 5, n = 4 | A1 | [4] |
**(i)** **(a)** $X$ or 5 | B1 | [1]
**(b)** $V$ or 3 | B1 | Should mention values or prob
| | Not just graph or spread eg 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]
**(ii)** $\frac{2 + 1}{2} \times 0.5$ or $\int_0^{0.5}(2 - 2x)dx$ | M1 | ('or' method requires linear function and correct limits)
= 0.75 | A1 | [2] | CWO
**(iii)** **(a)** $\int ax^n dx = 1$ | M1 | Attempt integ of correct form = 1 (ignore limits)
$$\left[\frac{ax^{n+1}}{n+1}\right]_0^1 = 1$$ | A1 | Correct integrand & limits
$$\frac{a}{n+1} = 1$$ | A1 | No errors seen
(a = n + 1 AG) | | [3]
**(b)** $\int ax^n dx = \frac{5}{6}$ oe | M1* | Integral of form $xf(x)hx = \frac{5}{6}$, ignore limits
$$\left[\frac{ax^{n+2}}{n+2}\right]_0^5 \text{ oe}$$ | A1 | Correct integrand & limits
$$\frac{a}{n+2} = \frac{5}{6}$$
(6a = 5n + 10) | M1dep | Attempt to use $a = n + 1$ within 2nd equ to get an equ in $n$ (or $a$)
a = 5, n = 4 | A1 | [4]7
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]{6e90d172-0292-4bb6-9d4a-506669076d4e-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]}}