| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Standard +0.3 This is a straightforward multi-part PDF question requiring standard techniques: (i) integration to find k, (ii) probability calculation via integration, (iii) solving for a quartile, and (iv) binomial probability. All parts follow routine procedures with no novel insight required. The integration of cos x is elementary, making this slightly easier than average for A-level statistics. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\int_0^{\pi/4} k\cos x \, dx = 1\) | M1 | Equating to 1 and attempt to integrate with limits |
| \([k\sin x]_0^{\pi/4} = 1\) | ||
| \(k\sin(\pi/4) = 1 \Rightarrow k/\sqrt{2} = 1\) | ||
| \(k = \sqrt{2}\) AG | A1 [2] | Correct answer legit obtained (no decimals seen) |
| (ii) \(\int_{0.4}^{\pi/4} k\cos x \, dx = [k\sin x]_{0.4}^{\pi/4}\) | M1 | Attempt to integrate from 0.4 to \(\pi/4\) o.e. |
| \(= 1 - k\sin(0.4) = 0.449\) | A1 [2] | Correct answer |
| (iii) \(\int_0^{Q3} k\cos x \, dx = 0.75\) | M1 | Equation with integral on one side and 0.75 on the other o.e. |
| \([k\sin x]_0^{Q3} = 0.75\) | M1 | Attempt to solve their integral for Q3 |
| \(k\sin Q3 - 0 = 0.75\) | ||
| \(Q3 = 0.559\) | A1 [3] | Correct answer |
| (iv) \(^5C_3 \times (0.25)^3 \times (0.75)^2\) | M1 | Binomial expression involving \(^5C_3\), 0.25 and 0.75 |
| \(= 0.0879 (45/512)\) | A1 [2] | Correct answer |
**(i)** $\int_0^{\pi/4} k\cos x \, dx = 1$ | M1 | Equating to 1 and attempt to integrate with limits
$[k\sin x]_0^{\pi/4} = 1$ |
$k\sin(\pi/4) = 1 \Rightarrow k/\sqrt{2} = 1$ |
$k = \sqrt{2}$ AG | A1 [2] | Correct answer legit obtained (no decimals seen)
**(ii)** $\int_{0.4}^{\pi/4} k\cos x \, dx = [k\sin x]_{0.4}^{\pi/4}$ | M1 | Attempt to integrate from 0.4 to $\pi/4$ o.e.
$= 1 - k\sin(0.4) = 0.449$ | A1 [2] | Correct answer
**(iii)** $\int_0^{Q3} k\cos x \, dx = 0.75$ | M1 | Equation with integral on one side and 0.75 on the other o.e.
$[k\sin x]_0^{Q3} = 0.75$ | M1 | Attempt to solve their integral for Q3
$k\sin Q3 - 0 = 0.75$ |
$Q3 = 0.559$ | A1 [3] | Correct answer
**(iv)** $^5C_3 \times (0.25)^3 \times (0.75)^2$ | M1 | Binomial expression involving $^5C_3$, 0.25 and 0.75
$= 0.0879 (45/512)$ | A1 [2] | Correct answer5 The continuous random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} k \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = \sqrt { } 2$.\\
(ii) Find $\mathrm { P } ( X > 0.4 )$.\\
(iii) Find the upper quartile of $X$.\\
(iv) Find the probability that exactly 3 out of 5 random observations of $X$ have values greater than the upper quartile.
\hfill \mbox{\textit{CAIE S2 2009 Q5 [9]}}