| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Explain why not valid PDF |
| Difficulty | Standard +0.3 This is a multi-part question on continuous probability distributions that requires understanding of variance/standard deviation and integration, but the calculations are straightforward. Part (i) requires visual interpretation of spread, part (ii)(a) is a standard variance calculation with given formula, part (ii)(b) is routine integration, and part (ii)(c) exploits symmetry. All techniques are standard S2 material with no novel problem-solving required. |
| 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 |
| \(\sigma_X,\ \sigma_Z,\ \sigma_Y,\ \sigma_W\) or \(X, Z, Y, W\) | B2 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean \(= 0\) stated or found, or "\(-0\)" seen | B1 | |
| \(\frac{1}{18}\int_{-3}^{3} x^4\,dx - 0\) | M1 | Attempt integral \(x^2 f(x)\). Ignore limits. Allow without "\(-0\)" |
| \(= \frac{1}{18}\left[\frac{x^5}{5}\right]_{-3}^{3} = \frac{1}{18}\left[\frac{3^5}{5} + \frac{3^5}{5}\right]\) oe \(= 5.4\) | ||
| \(\text{sd} = \sqrt{5.4}\) or \(\sqrt{\frac{1}{18}\left[\frac{3^5}{5} + \frac{3^5}{5}\right]}\) or \(2.324\) | A1 | Must see \(\sqrt{\text{correct expression}}\) or 5.4 or 2.324 or better |
| \(\text{sd} = 2.32\) (3 sf) AG | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{18}\int_{\text{'2.324'}}^{3} x^2\,dx\) | M1 | Attempt to integrate \(f(x)\), ignore limits |
| \(\frac{1}{18}\left[\frac{x^3}{3}\right]_{\text{'2.324'}}^{3} = \frac{1}{18}\left[\frac{3^3}{3} - \frac{\text{'2.324'}^3}{3}\right]\) | A1 | Sub correct limits into correct integral |
| \(= 0.268\) (3 sf) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0\) | B1 | [1] |
## Question 8:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sigma_X,\ \sigma_Z,\ \sigma_Y,\ \sigma_W$ or $X, Z, Y, W$ | B2 | [2] | B1 if two adjacent sds interchanged, ie $\sigma_Z, \sigma_X, \sigma_Y, \sigma_W$ or $\sigma_X, \sigma_Y, \sigma_Z, \sigma_W$ or $\sigma_X, \sigma_Z, \sigma_W, \sigma_Y$; B1 for correct order reversed |
### Part (ii)(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean $= 0$ stated or found, or "$-0$" seen | B1 | |
| $\frac{1}{18}\int_{-3}^{3} x^4\,dx - 0$ | M1 | Attempt integral $x^2 f(x)$. Ignore limits. Allow without "$-0$" |
| $= \frac{1}{18}\left[\frac{x^5}{5}\right]_{-3}^{3} = \frac{1}{18}\left[\frac{3^5}{5} + \frac{3^5}{5}\right]$ oe $= 5.4$ | | |
| $\text{sd} = \sqrt{5.4}$ or $\sqrt{\frac{1}{18}\left[\frac{3^5}{5} + \frac{3^5}{5}\right]}$ or $2.324$ | A1 | Must see $\sqrt{\text{correct expression}}$ or 5.4 or 2.324 or better |
| $\text{sd} = 2.32$ (3 sf) AG | | [3] |
### Part (ii)(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{18}\int_{\text{'2.324'}}^{3} x^2\,dx$ | M1 | Attempt to integrate $f(x)$, ignore limits |
| $\frac{1}{18}\left[\frac{x^3}{3}\right]_{\text{'2.324'}}^{3} = \frac{1}{18}\left[\frac{3^3}{3} - \frac{\text{'2.324'}^3}{3}\right]$ | A1 | Sub correct limits into correct integral |
| $= 0.268$ (3 sf) | A1 | [3] | Allow 0.269 |
### Part (ii)(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0$ | B1 | [1] |8\\
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The diagrams show the probability density functions of four random variables $W , X , Y$ and $Z$. Each of the four variables takes values between - 3 and 3 only, and their standard deviations are $\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }$ and $\sigma _ { Z }$ respectively.\\
(i) List $\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }$ and $\sigma _ { Z }$ in order of size, starting with the largest.\\
(ii) The probability density function of $X$ is given by
$$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\sigma _ { X } = 2.32$ correct to 3 significant figures.
\item Calculate $\mathrm { P } \left( X > \sigma _ { X } \right)$.
\item Write down the value of $\mathrm { P } \left( X > 2 \sigma _ { X } \right)$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2016 Q8 [9]}}