| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | PDF with multiple constants |
| Difficulty | Standard +0.3 This is a standard S2 PDF question requiring integration to find constants using the total probability condition, then finding the equation of a linear function, and finally using the expectation formula. All steps are routine applications of well-practiced techniques 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 |
|---|---|---|
| \(\frac{1}{2} \times a \times b = 1\) | M1 | Attempt \(\Delta\) area \(= 1\) or \(\int(b - bx/a)\,dx = 1\) with correct limits |
| \(b = \frac{2}{a}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{grad} = -\frac{2}{a^2}\) or \(-\frac{b}{a}\) | B1 | Allow without '\(-\)' sign (could be implied or seen in (i)) |
| \(y - \left(\frac{2}{a}\right) = \text{grad} \times x\) or \(y = \text{grad} \times (x-a)\) | M1 | Correct use of \(y = mx + c\) or \(y - y_1 = m(x-x_1)\) with \((0,b)\) or \((a,0)\) including attempt at substitution of their \(b\) |
| \(y - \left(\frac{2}{a}\right) = -\frac{2}{a^2}x\) or \(y = -\frac{2}{a^2}(x-a)\) and \(y = \frac{2}{a} - \frac{2}{a^2}x\) | A1 | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^a \left(\frac{2}{a}x - \frac{2}{a^2}x^2\right)dx\) | M1 | Attempt int \(xf(x)\) ignore limits |
| \(= \left[\frac{1}{a}x^2 - \frac{2}{3a^2}x^3\right]_0^a\) | A1 | Correct integration ignore limits |
| \(a - \frac{2}{3}a = 0.5\) | M1 | Sub correct limits into their integral and \(= 0.5\) |
| \(a = 1.5\) | A1 |
## Question 5(i):
| $\frac{1}{2} \times a \times b = 1$ | M1 | Attempt $\Delta$ area $= 1$ or $\int(b - bx/a)\,dx = 1$ with correct limits |
| $b = \frac{2}{a}$ | A1 | |
---
## Question 5(ii):
| $\text{grad} = -\frac{2}{a^2}$ or $-\frac{b}{a}$ | B1 | Allow without '$-$' sign (could be implied or seen in (i)) |
| $y - \left(\frac{2}{a}\right) = \text{grad} \times x$ or $y = \text{grad} \times (x-a)$ | M1 | Correct use of $y = mx + c$ or $y - y_1 = m(x-x_1)$ with $(0,b)$ or $(a,0)$ including attempt at substitution of their $b$ |
| $y - \left(\frac{2}{a}\right) = -\frac{2}{a^2}x$ or $y = -\frac{2}{a^2}(x-a)$ and $y = \frac{2}{a} - \frac{2}{a^2}x$ | A1 | No errors seen |
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## Question 5(iii):
| $\int_0^a \left(\frac{2}{a}x - \frac{2}{a^2}x^2\right)dx$ | M1 | Attempt int $xf(x)$ ignore limits |
| $= \left[\frac{1}{a}x^2 - \frac{2}{3a^2}x^3\right]_0^a$ | A1 | Correct integration ignore limits |
| $a - \frac{2}{3}a = 0.5$ | M1 | Sub correct limits into their integral and $= 0.5$ |
| $a = 1.5$ | A1 | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{b054d0a0-01b6-4785-807c-851551b90544-06_382_743_260_699}
The diagram shows the probability density function, f , of a random variable $X$, in terms of the constants $a$ and $b$.\\
(i) Find $b$ in terms of $a$.\\
(ii) Show that $\mathrm { f } ( x ) = \frac { 2 } { a } - \frac { 2 } { a ^ { 2 } } x$.\\
(iii) Given that $\mathrm { E } ( X ) = 0.5$, find $a$.\\
\hfill \mbox{\textit{CAIE S2 2018 Q5 [9]}}