| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Direct variance calculation from pdf |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring standard pdf techniques: using the total probability property to find the constant, calculating E(X) and Var(X) using integration, and solving a cumulative probability equation. All steps are routine applications of formulas with no conceptual challenges or novel problem-solving required. Slightly easier than average due to the simple uniform distribution in part (a) and straightforward integration in part (b). |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.5 \times \frac{1}{a} = \left(\frac{0.5}{a}\right)\) | M1 | Or attempt to integrate \(f(x)\) (\(=1/a\)) between 0 and 0.5 |
| \(= \frac{1}{2a}\) oe | A1 | Accept \(0.5/a\) for A1 |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{a}{2}\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_0^a \frac{x^2}{a}\,dx - \left('\frac{a}{2}'\right)^2\) | M1 | Integrate \(x^2 f(x)\) from 0 to \(a\) and sub their mean\(^2\) |
| \(\text{Var}(X) = \frac{a^2}{3} - \frac{a^2}{4}\) | A1 | Must see this line oe |
| \(\left(\text{Var}(X) = \frac{a^2}{12}\right)\) AG | ||
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_2^b \frac{3}{2(t-1)^2}\,dt\) | M1 | Attempt to integrate \(g(t)\), ignore limits |
| \(\left[-\frac{3}{2(t-1)}\right]_2^b\) | A1 | Correct integral |
| \(-\frac{3}{2}\left(\frac{1}{(b-1)}-1\right) = \frac{3}{4}\); \(\left(1 - \frac{1}{(b-1)} = \frac{1}{2}\right)\) | M1 | Attempt subst correct limits in their integral and \(= \frac{3}{4}\) |
| \(b = 3\) | A1 | |
| Total: 4 |
## Question 4(a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.5 \times \frac{1}{a} = \left(\frac{0.5}{a}\right)$ | M1 | Or attempt to integrate $f(x)$ ($=1/a$) between 0 and 0.5 |
| $= \frac{1}{2a}$ oe | A1 | Accept $0.5/a$ for A1 |
| **Total: 2** | | |
---
## Question 4(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{a}{2}$ | B1 | |
| **Total: 1** | | |
---
## Question 4(a)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_0^a \frac{x^2}{a}\,dx - \left('\frac{a}{2}'\right)^2$ | M1 | Integrate $x^2 f(x)$ from 0 to $a$ and sub their mean$^2$ |
| $\text{Var}(X) = \frac{a^2}{3} - \frac{a^2}{4}$ | A1 | Must see this line oe |
| $\left(\text{Var}(X) = \frac{a^2}{12}\right)$ **AG** | | |
| **Total: 2** | | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_2^b \frac{3}{2(t-1)^2}\,dt$ | M1 | Attempt to integrate $g(t)$, ignore limits |
| $\left[-\frac{3}{2(t-1)}\right]_2^b$ | A1 | Correct integral |
| $-\frac{3}{2}\left(\frac{1}{(b-1)}-1\right) = \frac{3}{4}$; $\left(1 - \frac{1}{(b-1)} = \frac{1}{2}\right)$ | M1 | Attempt subst correct limits in their integral and $= \frac{3}{4}$ |
| $b = 3$ | A1 | |
| **Total: 4** | | |
---4
\begin{enumerate}[label=(\alph*)]
\item \\
\begin{tikzpicture}[>=Stealth]
% Coordinates
\coordinate (origin) at (0,0);
\coordinate (a_axis) at (4,0);
\coordinate (x_end) at (5,0);
\coordinate (y_end) at (0,3);
\coordinate (top_left) at (0,2);
\coordinate (top_right) at (4,2);
% Axes
\draw[->] (-0.7,0) -- (x_end) node[below right] {$x$};
\draw[->] (0,-0) -- (y_end) node[left] {$f(x)$};
% PDF rectangle (straight line parallel to x-axis from x=0 to x=a)
\draw[thick] (top_left) -- (top_right);
\draw[thick] (4.5, 0) -- (a_axis);
\draw[thick, dashed] (a_axis) -- (top_right);
\draw[thick] (-0.5, 0) -- (0,0);
\draw (0,0) -- (top_left);
% Labels
\node[below] at (origin) {$0$};
\node[below] at (a_axis) {$a$};
\end{tikzpicture}
The diagram shows the graph of the probability density function, f , of a random variable $X$, where $a$ is a constant greater than 0.5 . The graph between $x = 0$ and $x = a$ is a straight line parallel to the $x$-axis.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { P } ( X < 0.5 )$ in terms of $a$.
\item Find $\mathrm { E } ( X )$ in terms of $a$.
\item Show that $\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }$.
\end{enumerate}\item A random variable $T$ has probability density function given by
$$\operatorname { g } ( t ) = \begin{cases} \frac { 3 } { 2 ( t - 1 ) ^ { 2 } } & 2 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
Find the value of $b$ such that $\mathrm { P } ( T \leqslant b ) = \frac { 3 } { 4 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2019 Q4 [9]}}