| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from median |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on continuous probability distributions requiring standard techniques: (i) uses the normalization condition for a pdf, (ii) applies the median definition (integral = 0.5), and (iii) calculates expectation. All steps are routine applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a\int_1^b \frac{1}{x^2}\,dx = 1\) | M1 | Attempt int \(f(x)\) and \(= 1\), ignore limits |
| \(a\left[-\frac{1}{x}\right]_1^b = 1\) | A1 | correct integ and limits \(= 1\) |
| \(a\left[1 - \frac{1}{b}\right] = 1\) or \(a \times \frac{b-1}{b} = 1\); \(b = \frac{a}{a-1}\) AG | A1 | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a\int_1^{\frac{3}{2}} \frac{1}{x^2}\,dx = \frac{1}{2}\); \(a\left[-\frac{1}{x}\right]_1^{\frac{3}{2}} = \frac{1}{2}\) | M1 | Attempt int \(f(x)\) with limits 1 to \(\frac{3}{2}\) and \(= \frac{1}{2}\) |
| \(a\left[1 - \frac{2}{3}\right] = \frac{1}{2}\) | A1 | oe correct eqn in \(a\) |
| \(a = \frac{3}{2},\ b = 3\) | A1 | Both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{3}{2}\int_1^3 \frac{1}{x}\,dx\) | M1 | Attempt int \(xf(x)\), ignore limits – condone missing \(a\) |
| \(= \frac{3}{2}\left[\ln x\right]_1^3\) | A1 | FT Correct integ and *their* limits 1 to \(b\) – condone missing \(a\) |
| \(= \frac{3}{2}\ln 3\) or 1.65 (3 sf) | A1 | FT *their* \(a\) and \(b\) (valid \(b\) i.e. \(>1\)) |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a\int_1^b \frac{1}{x^2}\,dx = 1$ | M1 | Attempt int $f(x)$ and $= 1$, ignore limits |
| $a\left[-\frac{1}{x}\right]_1^b = 1$ | A1 | correct integ and limits $= 1$ |
| $a\left[1 - \frac{1}{b}\right] = 1$ or $a \times \frac{b-1}{b} = 1$; $b = \frac{a}{a-1}$ **AG** | A1 | No errors seen |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a\int_1^{\frac{3}{2}} \frac{1}{x^2}\,dx = \frac{1}{2}$; $a\left[-\frac{1}{x}\right]_1^{\frac{3}{2}} = \frac{1}{2}$ | M1 | Attempt int $f(x)$ with limits 1 to $\frac{3}{2}$ and $= \frac{1}{2}$ |
| $a\left[1 - \frac{2}{3}\right] = \frac{1}{2}$ | A1 | oe correct eqn in $a$ |
| $a = \frac{3}{2},\ b = 3$ | A1 | Both |
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3}{2}\int_1^3 \frac{1}{x}\,dx$ | M1 | Attempt int $xf(x)$, ignore limits – condone missing $a$ |
| $= \frac{3}{2}\left[\ln x\right]_1^3$ | A1 | FT Correct integ and *their* limits 1 to $b$ – condone missing $a$ |
| $= \frac{3}{2}\ln 3$ or 1.65 (3 sf) | A1 | FT *their* $a$ and $b$ (valid $b$ i.e. $>1$) |(i) Show that $b = \frac { a } { a - 1 }$.\\
(ii) Given that the median of $X$ is $\frac { 3 } { 2 }$, find the values of $a$ and $b$.\\
(iii) Use your values of $a$ and $b$ from part (ii) to find $\mathrm { E } ( X )$.\\
\hfill \mbox{\textit{CAIE S2 2019 Q6 [9]}}