| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find median or percentiles |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring integration of a linear pdf. Students must use the property that total area = 1 and P(X<1) = 0.25 to find parameters, then find the median by solving for when cumulative probability = 0.5. All steps are standard S2 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.5 \times 1 \times h = 0.25\), \(h = 0.5\), \(\text{grad} = 0.5\) | M1 | \(P(X < 2) = 4 \times P(X < 1)\) |
| \(f(x) = 0.5x\) | A1 | \(P(X < 2) = 1\), \(a = 2\) |
| \(0.5 \times a \times 0.5a = 1\) | M1 | \(0.5 \times 2 \times h' = 1\), \(h' = 1\) |
| \(a = 2\) | A1 | \(\text{grad} = 0.5\) |
| \(P(X < 2) = 1\) | A1 | \(f(x) = 0.5x\) |
| Total: | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^m 0.5x\, dx = 0.5\) | M1 | Attempt \(\int f(x)\,dx = 0.5\), ignore limits |
| \(= \left[\frac{x^2}{4}\right]_0^m = 0.5\) | A1FT | Correct integration (ft \(f(x)\)) & limits \(= 0.5\) |
| \(m = \sqrt{2}\) or \(1.41\) (3 sf) | A1 | Or by similarity \(m = \frac{1}{\sqrt{2}} \times 2\), \(= \sqrt{2}\) |
| Total: | 3 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.5 \times 1 \times h = 0.25$, $h = 0.5$, $\text{grad} = 0.5$ | M1 | $P(X < 2) = 4 \times P(X < 1)$ |
| $f(x) = 0.5x$ | A1 | $P(X < 2) = 1$, $a = 2$ |
| $0.5 \times a \times 0.5a = 1$ | M1 | $0.5 \times 2 \times h' = 1$, $h' = 1$ |
| $a = 2$ | A1 | $\text{grad} = 0.5$ |
| $P(X < 2) = 1$ | A1 | $f(x) = 0.5x$ |
| **Total:** | **5** | |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^m 0.5x\, dx = 0.5$ | M1 | Attempt $\int f(x)\,dx = 0.5$, ignore limits |
| $= \left[\frac{x^2}{4}\right]_0^m = 0.5$ | A1FT | Correct integration (ft $f(x)$) & limits $= 0.5$ |
| $m = \sqrt{2}$ or $1.41$ (3 sf) | A1 | Or by similarity $m = \frac{1}{\sqrt{2}} \times 2$, $= \sqrt{2}$ |
| **Total:** | **3** | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{c06524f0-a981-48a6-9af0-c4a3474396b3-06_394_723_258_705}
The diagram shows the graph of the probability density function, f , of a random variable $X$ which takes values between 0 and $a$ only. It is given that $\mathrm { P } ( X < 1 ) = 0.25$.\\
(i) Find, in any order,
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( X < 2 )$,
\item the value of $a$,
\item $\mathrm { f } ( x )$.\\
(ii) Find the median of $X$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2017 Q5 [8]}}