| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Moderate -0.3 This is a straightforward continuous probability distribution question requiring standard techniques: integrating the pdf to find k, then computing E(X) and Var(X) using standard formulas. The integration is routine (power rule), and all steps are textbook exercises with no problem-solving insight needed. Slightly easier than average due to the simple algebraic form. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\int_1^{\infty} \frac{k}{x^2} dx = 1\) | M1 | Attempt integ f(x) & "= 1"; ignore limits |
| \(\left[-\frac{k}{3x^3}\right]_1^{\infty} = 1 \infty\) | Correct integrand & limits leading to AG, no errors seen | |
| \((0 + \frac{k}{3} = 1 \Rightarrow k = 3\) AG) | A1 [2] | |
| (ii) \(\int_1^{\infty} x \times \frac{3}{x^4} dx\) | M1 | Attempt integ xf(x); ignore limits. |
| \(\left[-\frac{3}{2x^2}\right]_1^{\infty}\) | ||
| \(= \frac{3}{2}\) | A1 | CWO |
| \(\int_1^{\infty} x^2 \times \frac{3}{x^4} dx\) | M1* | Attempt integ \(x^2f(x)\); ignore limits. |
| \(\left[-\frac{3}{x}\right]_1^{\infty} (= 3)\) | A1 | Correct integrand; correct limits |
| "3" \(= \left(3 \times \frac{\alpha}{2}\right)\) | M1*dep | dep 2nd M1 attempt E(\(X^2\)) – [E(\(X\))]² |
| \(= \frac{3}{4}\) | A1 | cwo |
| [6] |
**(i)** $\int_1^{\infty} \frac{k}{x^2} dx = 1$ | M1 | Attempt integ f(x) & "= 1"; ignore limits
$\left[-\frac{k}{3x^3}\right]_1^{\infty} = 1 \infty$ | | Correct integrand & limits leading to AG, no errors seen
$(0 + \frac{k}{3} = 1 \Rightarrow k = 3$ **AG)** | A1 [2] |
**(ii)** $\int_1^{\infty} x \times \frac{3}{x^4} dx$ | M1 | Attempt integ xf(x); ignore limits.
$\left[-\frac{3}{2x^2}\right]_1^{\infty}$ | |
$= \frac{3}{2}$ | A1 | CWO
$\int_1^{\infty} x^2 \times \frac{3}{x^4} dx$ | M1* | Attempt integ $x^2f(x)$; ignore limits.
$\left[-\frac{3}{x}\right]_1^{\infty} (= 3)$ | A1 | Correct integrand; correct limits
"3" $= \left(3 \times \frac{\alpha}{2}\right)$ | M1*dep | dep 2nd M1 attempt E($X^2$) – [E($X$)]²
$= \frac{3}{4}$ | A1 | cwo
| [6] |5 The time, in minutes, taken by volunteers to complete a task is modelled by the random variable $X$ with probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 4 } } & x \geqslant 1 \\ 0 & \text { otherwise. } \end{cases}$$
(i) Show that $k = 3$.\\
(ii) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{CAIE S2 2010 Q5 [8]}}