| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | November |
| 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 standard S2 probability density function question requiring routine integration to find k, recognizing symmetry for E(X), and computing a probability. All techniques are textbook exercises 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 pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(k\int_{-2}^{2}(4 - x^2)\,dx = 1\) | M1 | Attempt integral \(f(x) = 1\), ignore limits |
| \(k\left[4x - \frac{x^3}{3}\right]_{-2}^{2} = 1\) | A1 | Correct integration & limits |
| \(k \times \frac{32}{3} = 1\) oe; not e.g. \(k \times 10.7 = k\) | ||
| \(k = \frac{3}{32}\) AG | A1 [3] | Exact answer correctly found |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Inverted parabola, vertex on \(y\) axis | B1 | Parabola must finish on \(x\) axis at \(\pm 2\), labelled (ignore markings on \(y\) axis) |
| \(E(X) = 0\) | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{3}{32}\int_{-2}^{1}(4 - x^2)\,dx\) | M1 | or \(1 - \frac{3}{32}\int_{1}^{2}(4 - x^2)\,dx\), ignore limits |
| \(\frac{3}{32}\left[4x - \frac{x^3}{3}\right]_{-2}^{1}\) | A1 | or \(1 - \frac{3}{32}\left[4x - \frac{x^3}{3}\right]_{1}^{2}\); correct integration and correct limits \(= 1 - \frac{3}{32}(8 - \frac{8}{3} - (4 - \frac{1}{3}))\) |
| \(= \frac{27}{32}\) or \(0.844\) (3 sf) | A1 [3] |
# Question 4:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $k\int_{-2}^{2}(4 - x^2)\,dx = 1$ | M1 | Attempt integral $f(x) = 1$, ignore limits |
| $k\left[4x - \frac{x^3}{3}\right]_{-2}^{2} = 1$ | A1 | Correct integration & limits |
| $k \times \frac{32}{3} = 1$ oe; not e.g. $k \times 10.7 = k$ | | |
| $k = \frac{3}{32}$ AG | A1 [3] | Exact answer correctly found |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Inverted parabola, vertex on $y$ axis | B1 | Parabola must finish on $x$ axis at $\pm 2$, labelled (ignore markings on $y$ axis) |
| $E(X) = 0$ | B1 [2] | |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{3}{32}\int_{-2}^{1}(4 - x^2)\,dx$ | M1 | or $1 - \frac{3}{32}\int_{1}^{2}(4 - x^2)\,dx$, ignore limits |
| $\frac{3}{32}\left[4x - \frac{x^3}{3}\right]_{-2}^{1}$ | A1 | or $1 - \frac{3}{32}\left[4x - \frac{x^3}{3}\right]_{1}^{2}$; correct integration and correct limits $= 1 - \frac{3}{32}(8 - \frac{8}{3} - (4 - \frac{1}{3}))$ |
| $= \frac{27}{32}$ or $0.844$ (3 sf) | A1 [3] | |
---4 A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} k \left( 4 - x ^ { 2 } \right) & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = \frac { 3 } { 32 }$.\\
(ii) Sketch the graph of $y = \mathrm { f } ( x )$ and hence write down the value of $\mathrm { E } ( X )$.\\
(iii) Find $\mathrm { P } ( X < 1 )$.
\hfill \mbox{\textit{CAIE S2 2015 Q4 [8]}}