| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Parameter interpretation in context |
| Difficulty | Standard +0.3 This is a straightforward probability density function question requiring standard techniques: integration to find k using the total probability condition, contextual interpretation of a parameter, and solving P(X ≤ 0.5) = 0.75. Part (i) is routine integration, part (ii) is simple interpretation (maximum value), and part (iii) involves solving a simple equation. All steps are standard S2 material with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^a \frac{k}{(x+1)^2} dx = 1\) | M1 | Any attempt to integrate \(f(x)\) and \(= 1\). Ignore limits |
| \(-\left[\frac{k}{(x+1)}\right]_0^a = 1\) | M1 | Attempt to substitute correct limits into correct integral |
| \(-k\left(\frac{1}{a+1} - 1\right) = 1\) | ||
| \(k \times \frac{a}{a+1} = 1\) and \(k = \frac{a+1}{a}\) AG | A1 | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Max time allowed by model (for runners to finish) | B1 | Allow: All runners finish in time \(a\) or less or Longest time (taken by any runner) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{a+1}{a} \int_0^{0.5} \frac{1}{(x+1)^2} dx = \frac{3}{4}\) | M1 | Attempt to integrate \(f(x)\) and \(= \frac{3}{4}\); ignore limits. Condone missing/incorrect \(k\) |
| \(-\frac{a+1}{a}\left[\frac{1}{(x+1)}\right]_0^{0.5} = \frac{3}{4}\) | M1 | Attempt to substitute correct limits into correct integral. Condone missing/incorrect \(k\) |
| \(-\frac{a+1}{a}\left(\frac{2}{3} - 1\right) = \frac{3}{4}\) | ||
| \(a = 0.8\) | A1 |
## Question 4(i):
| $\int_0^a \frac{k}{(x+1)^2} dx = 1$ | M1 | Any attempt to integrate $f(x)$ and $= 1$. Ignore limits |
|---|---|---|
| $-\left[\frac{k}{(x+1)}\right]_0^a = 1$ | M1 | Attempt to substitute correct limits into correct integral |
| $-k\left(\frac{1}{a+1} - 1\right) = 1$ | | |
| $k \times \frac{a}{a+1} = 1$ and $k = \frac{a+1}{a}$ AG | A1 | No errors seen |
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## Question 4(ii):
| Max time allowed by model (for runners to finish) | B1 | Allow: All runners finish in time $a$ or less or Longest time (taken by any runner) |
---
## Question 4(iii):
| $\frac{a+1}{a} \int_0^{0.5} \frac{1}{(x+1)^2} dx = \frac{3}{4}$ | M1 | Attempt to integrate $f(x)$ and $= \frac{3}{4}$; ignore limits. Condone missing/incorrect $k$ |
|---|---|---|
| $-\frac{a+1}{a}\left[\frac{1}{(x+1)}\right]_0^{0.5} = \frac{3}{4}$ | M1 | Attempt to substitute correct limits into correct integral. Condone missing/incorrect $k$ |
| $-\frac{a+1}{a}\left(\frac{2}{3} - 1\right) = \frac{3}{4}$ | | |
| $a = 0.8$ | A1 | |
---4 The time, $X$ hours, taken by a large number of runners to complete a race is modelled by the probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $k$ and $a$ are constants.\\
(i) Show that $k = \frac { a + 1 } { a }$.\\
(ii) State what the constant $a$ represents in this context.\\
Three quarters of the runners take half an hour or less to complete the race.\\
(iii) Find the value of $a$.\\
\hfill \mbox{\textit{CAIE S2 2018 Q4 [7]}}