| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Expectation of function of X |
| Difficulty | Standard +0.8 This Further Maths question requires finding the pdf from the cdf (straightforward differentiation), computing E(√X) and Var(√X) using integration (non-standard functions of X requiring careful setup), and deriving a new pdf through transformation of variables using the Jacobian method. While each component uses standard techniques, the combination of multiple non-trivial integrations and the transformation in part (c) elevates this above routine exercises. |
| Spec | 5.03c Calculate mean/variance: by integration5.03g Cdf of transformed variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^9 \frac{2}{81} x^{1.5}\, dx\) | B1 | \(f(x) = \frac{2}{81}x\), correct expression as integrand |
| \(\frac{2}{81} \times \frac{2}{5} x^{2.5}\) | M1 | Integrate, FT only on their PDF expression |
| \(\frac{12}{5} = 2.4\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^9 \frac{2}{81} x^2\, dx - (\text{their } a)^2\) | M1 | |
| \(\frac{2}{81} \times \frac{9^3}{3} - 2.4^2 = \frac{6}{25}\) | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G(y) = \frac{1}{81}y^6\) | M1 | |
| \(g(y) = \frac{2}{27}y^5\) | M1 | |
| Fully correct including 'for \(0 \leq y \leq \sqrt[3]{9}\) and 0 otherwise' | A1 | Condone 2.08 |
| Total | 3 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^9 \frac{2}{81} x^{1.5}\, dx$ | B1 | $f(x) = \frac{2}{81}x$, correct expression as integrand |
| $\frac{2}{81} \times \frac{2}{5} x^{2.5}$ | M1 | Integrate, FT only on their PDF expression |
| $\frac{12}{5} = 2.4$ | A1 | |
| **Total** | **3** | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^9 \frac{2}{81} x^2\, dx - (\text{their } a)^2$ | M1 | |
| $\frac{2}{81} \times \frac{9^3}{3} - 2.4^2 = \frac{6}{25}$ | A1 | |
| **Total** | **2** | |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G(y) = \frac{1}{81}y^6$ | M1 | |
| $g(y) = \frac{2}{27}y^5$ | M1 | |
| Fully correct including 'for $0 \leq y \leq \sqrt[3]{9}$ and 0 otherwise' | A1 | Condone 2.08 |
| **Total** | **3** | |
---3 The continuous random variable $X$ has cumulative distribution function F given by
$$F ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 81 } x ^ { 2 } & 0 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { E } ( \sqrt { X } )$.
\item Find $\operatorname { Var } ( \sqrt { X } )$.
\item The random variable $Y$ is given by $Y ^ { 3 } = X$. Find the probability density function of $Y$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q3 [8]}}