| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Calculate probabilities and expectations |
| Difficulty | Moderate -0.3 This is a straightforward continuous uniform distribution question requiring standard techniques: identifying the median by symmetry, calculating probabilities using areas, deriving variance using the standard formula, and manipulating probabilities algebraically. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{a}{2}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{4}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f(x) = \frac{1}{a}\) | B1 | SOI (may be seen in part (a) or part (b)) |
| \(E(X) = \frac{a}{2}\) | B1 | SOI |
| \(\int_0^a \frac{1}{a} x^2 \, dx\) | M1 | Attempt integrate *their* \(f(x) \times x^2\) with correct limits |
| \(= \left[\frac{x^3}{3a}\right]_0^a = \frac{a^2}{3}\) | A1 | |
| \(\frac{a^2}{3} - \left(\frac{a}{2}\right)^2\) or \(\frac{a^2}{3} - \frac{a^2}{4} = \frac{a^2}{12}\) AG | A1 | Must see previous line and answer; No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P\!\left(X < \frac{b}{3}\right) = \frac{p}{3}\) | M1 | SOI (could be on a diagram); OR by integration: \(\text{prob} = 1 - \frac{2}{3}\cdot\frac{b}{a}\) |
| \(P\!\left(\frac{b}{3} < X < a - \frac{b}{3}\right) = 1 - \frac{2p}{3}\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{a}{2}$ | B1 | |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{4}$ | B1 | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $f(x) = \frac{1}{a}$ | B1 | SOI (may be seen in part (a) or part (b)) |
| $E(X) = \frac{a}{2}$ | B1 | SOI |
| $\int_0^a \frac{1}{a} x^2 \, dx$ | M1 | Attempt integrate *their* $f(x) \times x^2$ with correct limits |
| $= \left[\frac{x^3}{3a}\right]_0^a = \frac{a^2}{3}$ | A1 | |
| $\frac{a^2}{3} - \left(\frac{a}{2}\right)^2$ or $\frac{a^2}{3} - \frac{a^2}{4} = \frac{a^2}{12}$ **AG** | A1 | Must see previous line and answer; No errors seen |
---
## Question 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P\!\left(X < \frac{b}{3}\right) = \frac{p}{3}$ | M1 | SOI (could be on a diagram); OR by integration: $\text{prob} = 1 - \frac{2}{3}\cdot\frac{b}{a}$ |
| $P\!\left(\frac{b}{3} < X < a - \frac{b}{3}\right) = 1 - \frac{2p}{3}$ | A1 | |
---6 A random variable $X$ has probability density function f . The graph of $\mathrm { f } ( x )$ is a straight line segment parallel to the $x$-axis from $x = 0$ to $x = a$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item State, in terms of $a$, the median of $X$.
\item Find $\mathrm { P } \left( X > \frac { 3 } { 4 } a \right)$.
\item Show that $\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }$.
\item Given that $\mathrm { P } ( X < b ) = p$, where $0 < b < \frac { 1 } { 2 } a$, find $\mathrm { P } \left( \frac { 1 } { 3 } b < X < a - \frac { 1 } { 3 } b \right)$ in terms of $p$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q6 [9]}}