| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate probability P(X in interval) |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing basic continuous probability concepts: (a) requires simple integration of a polynomial pdf over an interval, (b)(i) uses the property that total probability equals 1 with semicircle area, (b)(ii) exploits symmetry for expectation, and (b)(iii) uses symmetry properties. All parts are standard textbook exercises requiring routine application of well-known techniques with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_0^{0.5}(1.5t - 0.75t^2)\,dt\) o.e. | M1 | Attempt \(\int f(t)\) |
| \(= \left[0.75t^2 - 0.25t^3\right]_0^{0.5}\) o.e. | A1 | Correct integration and limits |
| \(= \frac{5}{32}\) or \(0.156\) (3 sf) | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}\pi a^2 = 1\) or \(\pi a^2 = 2\) o.e. | M1 | Attempt to find the area and equate to 1 |
| \(a = \sqrt{\frac{2}{\pi}}\) or \(0.798\) (3 sf) | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0\) | B1 | Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Symmetry stated, seen or implied | M1 | Could be a diagram |
| \(0.8\) | A1 | As final answer. Total: 2 |
## Question 3(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_0^{0.5}(1.5t - 0.75t^2)\,dt$ o.e. | M1 | Attempt $\int f(t)$ |
| $= \left[0.75t^2 - 0.25t^3\right]_0^{0.5}$ o.e. | A1 | Correct integration and limits |
| $= \frac{5}{32}$ or $0.156$ (3 sf) | A1 | **Total: 3** |
## Question 3(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}\pi a^2 = 1$ or $\pi a^2 = 2$ o.e. | M1 | Attempt to find the area and equate to 1 |
| $a = \sqrt{\frac{2}{\pi}}$ or $0.798$ (3 sf) | A1 | **Total: 2** |
## Question 3(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0$ | B1 | **Total: 1** |
## Question 3(b)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Symmetry stated, seen or implied | M1 | Could be a diagram |
| $0.8$ | A1 | As final answer. **Total: 2** |
---3
\begin{enumerate}[label=(\alph*)]
\item The time for which Lucy has to wait at a certain traffic light each day is $T$ minutes, where $T$ has probability density function given by
$$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{f9436a68-ec88-4feb-9c06-fc29fe53d1fe-2_405_793_1471_715}
The diagram shows the graph of the probability density function, g , of a random variable $X$. The graph of g is a semicircle with centre $( 0,0 )$ and radius $a$. Elsewhere $\mathrm { g } ( x ) = 0$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $a$.
\item State the value of $\mathrm { E } ( X )$.
\item Given that $\mathrm { P } ( X < - c ) = 0.2$, find $\mathrm { P } ( X < c )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2014 Q3 [8]}}