| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from probability condition |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring (i) integration to find k using the total probability condition, and (ii) solving an equation involving the cumulative distribution function. Both parts use standard techniques with simple logarithmic integration, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_3^5 \frac{1}{x-1} dx = 1\) | M1 | Attempt integ f(x) & '= 1' ignore limits |
| \([\ln(x-1)]_3^5 = 1\) | A1 | Correctly integrated; ignore limits |
| \(k(\ln 4 - \ln 2) = 1\); \(k\ln 2 = 1\) \((k = \frac{1}{\ln 2}\) AG) | M1 A1 [4] | Subst of limits 3, 5; No errors seen. No decimals seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{\ln 2} \int_3^x \frac{1}{x-1}dx = 0.75\) | M1* | Attempt integ f(x), unknown limit, & '= 0.75'or '= 0.25' |
| \(\frac{1}{\ln 2}[\ln(x-1)]_3^x = 0.75\) | A1 | oe. Fully correct eqn after subst limits |
| \(\frac{1}{\ln 2}[(\ln(x-1) - \ln 2) = 0.75\); \(\ln(x-1) = (0.75 \times \ln 2 + \ln 2)\); \(\ln(x-1) = 1.75 \times \ln 2\) or \(x-1 = 3.36\); \(x = 4.36\) (3 sfs) | M1 dep* A1 [4] | oe. Correct manipulation of logs to find x |
**(i)**
$\int_3^5 \frac{1}{x-1} dx = 1$ | M1 | Attempt integ f(x) & '= 1' ignore limits
$[\ln(x-1)]_3^5 = 1$ | A1 | Correctly integrated; ignore limits
$k(\ln 4 - \ln 2) = 1$; $k\ln 2 = 1$ $(k = \frac{1}{\ln 2}$ **AG**) | M1 A1 [4] | Subst of limits 3, 5; No errors seen. No decimals seen
**(ii)**
$\frac{1}{\ln 2} \int_3^x \frac{1}{x-1}dx = 0.75$ | M1* | Attempt integ f(x), unknown limit, & '= 0.75'or '= 0.25'
$\frac{1}{\ln 2}[\ln(x-1)]_3^x = 0.75$ | A1 | oe. Fully correct eqn after subst limits
$\frac{1}{\ln 2}[(\ln(x-1) - \ln 2) = 0.75$; $\ln(x-1) = (0.75 \times \ln 2 + \ln 2)$; $\ln(x-1) = 1.75 \times \ln 2$ or $x-1 = 3.36$; $x = 4.36$ (3 sfs) | M1 dep* A1 [4] | oe. Correct manipulation of logs to find x5 A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { x - 1 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = \frac { 1 } { \ln 2 }$.\\
(ii) Find $a$ such that $\mathrm { P } ( X < a ) = 0.75$.
\hfill \mbox{\textit{CAIE S2 2012 Q5 [8]}}