| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from expectation |
| Difficulty | Standard +0.3 This is a straightforward probability density function question requiring integration to find constants using standard conditions (∫f(x)dx=1 and E(X)=3). The integration of x^(-1/2) and x^(1/2) is routine A-level calculus, and solving the resulting simultaneous equations is algebraic manipulation. Slightly easier than average as it follows a standard template with no conceptual surprises. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k\int_{0}^{a} \dfrac{1}{\sqrt{x}}\,dx = 1\) | M1 | Attempt int \(f(x)\) and \(= 1\), ignore limits |
| \(\left(2k[x^{0.5}]_{0}^{a} = 1\right)\); \(2ka^{0.5} = 1\) or \(a = \dfrac{1}{4k^2}\) | A1 | OE; a correct eqn in \(k\) & \(a\) after sub limits |
| \(k\int_{0}^{a} \dfrac{x}{\sqrt{x}}\,dx = 3\) | M1 | Attempt int \(xf(x)\) and \(= 3\) |
| e.g. \(\dfrac{2}{3}ka^{1.5} = 3\) or \(a^3 = \dfrac{81}{4k^2}\) | A1 | OE; a correct eqn in \(k\) and \(a\) after sub limits |
| e.g. \(a^2 = 81\) or e.g. \(k^2 = \dfrac{81}{4 \times 9^3}\) | M1 | Attempt eliminate one letter |
| \(a = 9\) | A1 | Convincingly obtained |
| e.g. \(k = \dfrac{9}{54}\); \(k = \dfrac{1}{6}\) AG | A1 | |
| Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{6}\int_0^m \frac{1}{\sqrt{x}}dx = 0.5\) OE | M1 | Attempt int \(f(x)\), unknown limit and \(= 0.5\) |
| \(\frac{1}{3}m^{0.5} = 0.5\) | A1 | a correct equation in \(m\) after sub limits |
| \(m = 2.25\) | A1 | |
| 3 |
## Question 4:
**Part 4(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k\int_{0}^{a} \dfrac{1}{\sqrt{x}}\,dx = 1$ | M1 | Attempt int $f(x)$ and $= 1$, ignore limits |
| $\left(2k[x^{0.5}]_{0}^{a} = 1\right)$; $2ka^{0.5} = 1$ or $a = \dfrac{1}{4k^2}$ | A1 | OE; a correct eqn in $k$ & $a$ after sub limits |
| $k\int_{0}^{a} \dfrac{x}{\sqrt{x}}\,dx = 3$ | M1 | Attempt int $xf(x)$ and $= 3$ |
| e.g. $\dfrac{2}{3}ka^{1.5} = 3$ or $a^3 = \dfrac{81}{4k^2}$ | A1 | OE; a correct eqn in $k$ and $a$ after sub limits |
| e.g. $a^2 = 81$ or e.g. $k^2 = \dfrac{81}{4 \times 9^3}$ | M1 | Attempt eliminate one letter |
| $a = 9$ | A1 | Convincingly obtained |
| e.g. $k = \dfrac{9}{54}$; $k = \dfrac{1}{6}$ AG | A1 | |
| | **Total: 7** | |
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{6}\int_0^m \frac{1}{\sqrt{x}}dx = 0.5$ OE | M1 | Attempt int $f(x)$, unknown limit and $= 0.5$ |
| $\frac{1}{3}m^{0.5} = 0.5$ | A1 | a correct equation in $m$ after sub limits |
| $m = 2.25$ | A1 | |
| | **3** | |
---4 The random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $k$ and $a$ are constants. It is given that $\mathrm { E } ( X ) = 3$.\\
(i) Find the value of $a$ and show that $k = \frac { 1 } { 6 }$.\\
(ii) Find the median of $X$.\\
\hfill \mbox{\textit{CAIE S2 2017 Q4 [10]}}