| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | PDF with multiple constants |
| Difficulty | Standard +0.3 This is a standard PDF question requiring integration to find a constant and calculate expectation. Part (a) uses the fundamental property that ∫f(x)dx = 1, and part (b) applies the definition E(X) = ∫xf(x)dx. Both integrations are straightforward using power rules, making this slightly easier than average for A-level statistics. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_{2}^{3}\left(\frac{a}{x^2} - \frac{18}{x^3}\right)dx = 1\) | M1 | Attempt integrate f(x), ignore limits and '= 1' |
| \(\left[-\frac{a}{x} + \frac{9}{x^2}\right]_{2}^{3} = 1\) | A1 | OE Correct integration and limits |
| \(\left[-\frac{a}{3} + 1 + \frac{a}{2} - \frac{9}{4}\right] = 1 \quad a = \frac{27}{2}\) (AG) | A1 | Must see correct substitution of limits. Correct working no errors seen |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_{2}^{3}\left(\frac{27}{2x} - \frac{18}{x^2}\right)dx\) | M1 | Attempt to integrate xf(x), ignore limits |
| \(\left[\frac{27}{2}\ln x + \frac{18}{x}\right]_{2}^{3}\) or \(\left[\frac{27}{2}\ln 2x + \frac{18}{x}\right]_{2}^{3}\) | A1 | Correct integration and limits. OE e.g. using ln 2x |
| \(= \frac{27}{2}\ln 3 + 6 - \frac{27}{2}\ln 2 - 9 = \frac{27}{2}\ln\frac{3}{2} - 3\) AG | A1 | Must see correct substitution of limits. Correct working no errors seen |
| Total: 3 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_{2}^{3}\left(\frac{a}{x^2} - \frac{18}{x^3}\right)dx = 1$ | M1 | Attempt integrate f(x), ignore limits and '= 1' |
| $\left[-\frac{a}{x} + \frac{9}{x^2}\right]_{2}^{3} = 1$ | A1 | OE Correct integration and limits |
| $\left[-\frac{a}{3} + 1 + \frac{a}{2} - \frac{9}{4}\right] = 1 \quad a = \frac{27}{2}$ (AG) | A1 | Must see correct substitution of limits. Correct working no errors seen |
| **Total: 3** | | |
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_{2}^{3}\left(\frac{27}{2x} - \frac{18}{x^2}\right)dx$ | M1 | Attempt to integrate xf(x), ignore limits |
| $\left[\frac{27}{2}\ln x + \frac{18}{x}\right]_{2}^{3}$ or $\left[\frac{27}{2}\ln 2x + \frac{18}{x}\right]_{2}^{3}$ | A1 | Correct integration and limits. OE e.g. using ln 2x |
| $= \frac{27}{2}\ln 3 + 6 - \frac{27}{2}\ln 2 - 9 = \frac{27}{2}\ln\frac{3}{2} - 3$ AG | A1 | Must see correct substitution of limits. Correct working no errors seen |
| **Total: 3** | | |4 A random variable $X$ has probability density function f defined by
$$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } - \frac { 18 } { x ^ { 3 } } & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \frac { 27 } { 2 }$.
\item Show that $\mathrm { E } ( X ) = \frac { 27 } { 2 } \ln \frac { 3 } { 2 } - 3$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q4 [6]}}