| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | PDF to CDF derivation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths statistics question requiring standard techniques: (a) integrate the PDF to find CDF using power rule, (b) apply the transformation formula for PDFs with a simple power transformation, (c) solve F(y)=0.5. All steps are routine applications of A-level Further Maths methods with no novel problem-solving 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 percentiles5.07a Non-parametric tests: when to use |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(F(x) = \frac{1}{6}\left[\frac{3}{2}x^{\frac{2}{3}} - 3x^{\frac{1}{3}}\right] + c\) | M1 | Integrate \(f(x)\), \(+c\) not required, at least one power correct |
| \(\frac{1}{4}x^{\frac{2}{3}} - \frac{1}{2}x^{\frac{1}{3}} + \frac{1}{4}\) | A1 | AEF |
| Correct ranges including \(1 \leqslant x \leqslant 27\) associated with their \(F(x)\); \(F(x) = 0\) for \(x < 1\) and \(F(x) = 1\) for \(x > 27\) | A1 | No gaps |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G(y) = \frac{1}{4}(y^2 - 2y + 1)\) | M1 | Using \(y = x^{\frac{1}{3}}\) in *their* \(F(x)\) |
| \(g(y) = \frac{1}{2}(y-1)\) | A1 | Correct, AEF, 0 otherwise not required |
| For \(1 \leqslant y \leqslant 3\) | B1 | Seen anywhere, correct variable |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{4}(y^2 - 2y + 1) = \frac{1}{2}\), \(\quad (y-1)^2 = 2\) OE | M1 | Equate their \(G(y)\) to \(\frac{1}{2}\) and solve to find \(y\) |
| \([y =] \ \ 1 + \sqrt{2}\) | A1 | Only |
| Total: 2 |
## Question 1:
### Part 1(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $F(x) = \frac{1}{6}\left[\frac{3}{2}x^{\frac{2}{3}} - 3x^{\frac{1}{3}}\right] + c$ | M1 | Integrate $f(x)$, $+c$ not required, at least one power correct |
| $\frac{1}{4}x^{\frac{2}{3}} - \frac{1}{2}x^{\frac{1}{3}} + \frac{1}{4}$ | A1 | AEF |
| Correct ranges including $1 \leqslant x \leqslant 27$ associated with their $F(x)$; $F(x) = 0$ for $x < 1$ and $F(x) = 1$ for $x > 27$ | A1 | No gaps |
| **Total: 3** | | |
### Part 1(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G(y) = \frac{1}{4}(y^2 - 2y + 1)$ | M1 | Using $y = x^{\frac{1}{3}}$ in *their* $F(x)$ |
| $g(y) = \frac{1}{2}(y-1)$ | A1 | Correct, AEF, 0 otherwise not required |
| For $1 \leqslant y \leqslant 3$ | B1 | Seen anywhere, correct variable |
| **Total: 3** | | |
### Part 1(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{4}(y^2 - 2y + 1) = \frac{1}{2}$, $\quad (y-1)^2 = 2$ OE | M1 | Equate their $G(y)$ to $\frac{1}{2}$ and solve to find $y$ |
| $[y =] \ \ 1 + \sqrt{2}$ | A1 | Only |
| **Total: 2** | | |1 The continuous random variable $X$ has probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find the cumulative distribution function of $X$.\\
The random variable $Y$ is defined by $Y = X ^ { \frac { 1 } { 3 } }$.
\item Find the probability density function of $Y$.
\item Find the exact value of the median of $Y$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q1 [8]}}