| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Geometric/graphical PDF with k |
| Difficulty | Standard +0.8 This question requires students to work with a geometric PDF (quarter circle), apply the fundamental property that total probability equals 1 (involving area of quarter circle = πa²/4), derive the PDF equation from the circle equation, and compute E(X) using integration by substitution or recognition of volume formulas. While the 'show that' format provides targets, the integration in part (c) requires either trigonometric substitution or geometric insight about volumes, making this significantly harder than routine PDF questions but still within S2 scope. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03d E(g(X)): general expectation formula |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{4}\pi a^2 = 1\) | M1 | OE. Attempt to set area \(= 1\). |
| \(a = \frac{2}{\sqrt{\pi}}\) | A1 | AG. Correct equation and correctly rearranged to \(a = \ldots\) No errors seen. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x^2 + y^2 = \left(\frac{2}{\sqrt{\pi}}\right)^2\) | M1 | Or \(x^2 + y^2 = a^2\). Must see at least one intermediate step. |
| \(f(x) = \sqrt{\frac{4}{\pi} - x^2}\) | A1 | AG. Convincingly rearranged to reach given answer. No errors seen. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_0^{\frac{2}{\sqrt{\pi}}} x\sqrt{\frac{4}{\pi} - x^2}\, dx\) | B1 | Correct expression for \(E(X) = \int x f(x)\, dx\) with correct limits (accept limits \(0\) and \(a\)). |
| \(-\frac{1}{3}\!\left[\left(\frac{4}{\pi} - x^2\right)^{\!\frac{3}{2}}\right]_0^{\frac{2}{\sqrt{\pi}}}\) | M1 | Integrate \(xf(x)\) with any limits or none. Must reach expression of form: any constant \(\times \left(\frac{4}{\pi} - x^2\right)^{\!\frac{3}{2}}\). |
| (wholly correct integration and limits) | A1 | |
| \(\frac{8}{3\sqrt{\pi^3}}\) | A1 | AG. Correctly obtained with no errors seen. |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{4}\pi a^2 = 1$ | M1 | OE. Attempt to set area $= 1$. |
| $a = \frac{2}{\sqrt{\pi}}$ | A1 | AG. Correct equation and correctly rearranged to $a = \ldots$ No errors seen. |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x^2 + y^2 = \left(\frac{2}{\sqrt{\pi}}\right)^2$ | M1 | Or $x^2 + y^2 = a^2$. Must see at least one intermediate step. |
| $f(x) = \sqrt{\frac{4}{\pi} - x^2}$ | A1 | AG. Convincingly rearranged to reach given answer. No errors seen. |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_0^{\frac{2}{\sqrt{\pi}}} x\sqrt{\frac{4}{\pi} - x^2}\, dx$ | B1 | Correct expression for $E(X) = \int x f(x)\, dx$ with correct limits (accept limits $0$ and $a$). |
| $-\frac{1}{3}\!\left[\left(\frac{4}{\pi} - x^2\right)^{\!\frac{3}{2}}\right]_0^{\frac{2}{\sqrt{\pi}}}$ | M1 | Integrate $xf(x)$ with any limits or none. Must reach expression of form: any constant $\times \left(\frac{4}{\pi} - x^2\right)^{\!\frac{3}{2}}$. |
| (wholly correct integration and limits) | A1 | |
| $\frac{8}{3\sqrt{\pi^3}}$ | A1 | AG. Correctly obtained with no errors seen. |6\\
\includegraphics[max width=\textwidth, alt={}, center]{5dfbc896-528c-40a6-a296-8a0aae90add4-10_451_469_255_799}
The diagram shows the graph of the probability density function, f , of a random variable $X$. The graph is a quarter circle entirely in the first quadrant with centre $( 0,0 )$ and radius $a$, where $a$ is a positive constant. Elsewhere $\mathrm { f } ( x ) = 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \frac { 2 } { \sqrt { \pi } }$.
\item Show that $\mathrm { f } ( x ) = \sqrt { \frac { 4 } { \pi } - x ^ { 2 } }$.
\item Show that $\mathrm { E } ( X ) = \frac { 8 } { 3 \sqrt { \pi ^ { 3 } } }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q6 [8]}}