| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Moderate -0.5 This is a straightforward continuous uniform distribution question requiring basic recall of pdf properties (area = 1) and the variance formula for uniform distributions. The algebra is simple: part (a) gives k = 1/a immediately, and part (b) requires substituting into Var(X) = a²/12 = 3 and solving. This is easier than average as it's a direct application of standard formulas with minimal problem-solving. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((k =)\ \frac{1}{a}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (Mean \(=\)) *their* \(k \times \frac{a^2}{2}\) \(\left(= \frac{a}{2}\right)\) | B1 FT | OE seen. FT *their* \(k\) |
| \(\frac{1}{a}\int_0^a x^2\,dx \left(= \frac{a^2}{3}\right)\) | M1 | Attempt at correct integral and use of limits. Accept in terms of \(k\) or incorrect \(k\) |
| \(-\left(\frac{a}{2}\right)^2 \left(= \frac{a^2}{12}\right)\) | M1 | For subtracting mean\(^2\), allow if integration not complete. FT incorrect values of \(k\) |
| \(\left(\frac{a^2}{12} = 3\right)\ a = 6\) | A1 | Can be in terms of \(k\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(k =)\ \frac{1}{a}$ | B1 | |
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| (Mean $=$) *their* $k \times \frac{a^2}{2}$ $\left(= \frac{a}{2}\right)$ | B1 FT | OE seen. FT *their* $k$ |
| $\frac{1}{a}\int_0^a x^2\,dx \left(= \frac{a^2}{3}\right)$ | M1 | Attempt at correct integral and use of limits. Accept in terms of $k$ or incorrect $k$ |
| $-\left(\frac{a}{2}\right)^2 \left(= \frac{a^2}{12}\right)$ | M1 | For subtracting mean$^2$, allow if integration not complete. FT incorrect values of $k$ |
| $\left(\frac{a^2}{12} = 3\right)\ a = 6$ | A1 | Can be in terms of $k$ |4\\
\includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-07_316_984_260_577}
The diagram shows the probability density function, $\mathrm { f } ( x )$, of a random variable $X$. For $0 \leqslant x \leqslant a$, $\mathrm { f } ( x ) = k$; elsewhere $\mathrm { f } ( x ) = 0$.
\begin{enumerate}[label=(\alph*)]
\item Express $k$ in terms of $a$.
\item Given that $\operatorname { Var } ( X ) = 3$, find $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q4 [5]}}