| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Standard +0.3 This is a standard S2 continuous probability distribution question requiring routine integration to find k, calculate E(X), and apply binomial probability. All steps follow textbook methods with no novel insight needed, making it slightly easier than average A-level difficulty. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(k\int_{1}^{2}(\frac{1}{x^2}+\frac{1}{x^3})\,dx = 1\) | M1 | Attempt integ \(f(x)\) & \(= 1\); ignore limits |
| \(k\left[-\frac{1}{x}-\frac{1}{2x^2}\right]_{1}^{2} = 1\) | A1 | Correct integral & limits & \(= 1\) |
| \(k\left[-\frac{1}{2}-\frac{1}{8}+1+\frac{1}{2}\right] = 1\) | A1 | Sufficient working must be shown, no errors seen |
| \(k = \frac{8}{7}\) AG | ||
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{8}{7}\int_{1}^{2}(\frac{1}{x}+\frac{1}{x^2})\,dx\) | M1 | Attempt integ \(xf(x)\), ignore limits |
| \(= \frac{8}{7}\left[\ln x - \frac{1}{x}\right]_{1}^{2}\) | A1 | Correct integral & limits, condone missing \(k\) |
| \(= \frac{8}{7}(\ln 2 + \frac{1}{2})\) or \(1.36\) (3 sf) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{8}{7}\int_{1}^{1.5}(\frac{1}{x^2}+\frac{1}{x^3})\,dx = \frac{8}{7}\left[-\frac{1}{x}-\frac{1}{2x^2}\right]_{1}^{1.5}\) | M1 | Attempt integration \(f(x)\) between 1 and 1.5 or between 1.5 and 2 |
| \(= \frac{44}{63}\) or \(0.698\ldots\) | A1 | Or \(\frac{19}{63}\) or \(0.302\) |
| \(\frac{44}{63}\cdot(1-\frac{44}{63})^2\) | M1 | FT their \(\frac{44}{63}\) |
| \(\times 3\) | M1 | Independent provided answer is \(<1\) |
| \(= 0.191\) | A1 | |
| 5 |
## Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $k\int_{1}^{2}(\frac{1}{x^2}+\frac{1}{x^3})\,dx = 1$ | M1 | Attempt integ $f(x)$ & $= 1$; ignore limits |
| $k\left[-\frac{1}{x}-\frac{1}{2x^2}\right]_{1}^{2} = 1$ | A1 | Correct integral & limits & $= 1$ |
| $k\left[-\frac{1}{2}-\frac{1}{8}+1+\frac{1}{2}\right] = 1$ | A1 | Sufficient working must be shown, no errors seen |
| $k = \frac{8}{7}$ **AG** | | |
| | **3** | |
## Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{8}{7}\int_{1}^{2}(\frac{1}{x}+\frac{1}{x^2})\,dx$ | M1 | Attempt integ $xf(x)$, ignore limits |
| $= \frac{8}{7}\left[\ln x - \frac{1}{x}\right]_{1}^{2}$ | A1 | Correct integral & limits, condone missing $k$ |
| $= \frac{8}{7}(\ln 2 + \frac{1}{2})$ or $1.36$ (3 sf) | A1 | |
| | **3** | |
## Question 7(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{8}{7}\int_{1}^{1.5}(\frac{1}{x^2}+\frac{1}{x^3})\,dx = \frac{8}{7}\left[-\frac{1}{x}-\frac{1}{2x^2}\right]_{1}^{1.5}$ | M1 | Attempt integration $f(x)$ between 1 and 1.5 or between 1.5 and 2 |
| $= \frac{44}{63}$ or $0.698\ldots$ | A1 | Or $\frac{19}{63}$ or $0.302$ |
| $\frac{44}{63}\cdot(1-\frac{44}{63})^2$ | M1 | FT their $\frac{44}{63}$ |
| $\times 3$ | M1 | Independent provided answer is $<1$ |
| $= 0.191$ | A1 | |
| | **5** | |7 A random variable $X$ has probability density function defined by
$$f ( x ) = \begin{cases} k \left( \frac { 1 } { x ^ { 2 } } + \frac { 1 } { x ^ { 3 } } \right) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = \frac { 8 } { 7 }$.\\
(ii) Find $\mathrm { E } ( X )$.\\
(iii) Three values of $X$ are chosen at random. Find the probability that one of these values is less than 1.5 and the other two are greater than 1.5.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE S2 2018 Q7 [11]}}