| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Calculate probabilities from CDF |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring standard CDF manipulations: converting P(X>a) to F(a), solving a quadratic equation, differentiating to find the PDF, and applying standard expectation/variance formulas. All techniques are routine for this module with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(\frac{1}{6}a(a+1) = 0.6\) | M1 | Putting \(F(x) = 0.6\) or \(1-0.4\) |
| \(a^2 + a - 3.6 = 0\) | ||
| \(a = \frac{-1 \pm \sqrt{1+4\times3.6}}{2}\) | M1 | Attempting completing the square or quadratic formula (one slip allowed); condone \(+\) instead of \(\pm\) |
| \(= 1.462...\) | ||
| \(a = 1.46\) only | A1 | Must reject other root if stated; condone awrt 1.46 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(f(x) = \frac{d}{dx}F(x) = \frac{1}{3}x + \frac{1}{6}\) | M1A1 | 1st M1: attempting to differentiate \(F(x)\), at least one \(x^n \to x^{n-1}\) |
| \(E(X) = \int_0^2 x\left(\frac{1}{3}x + \frac{1}{6}\right)dx\) | M1 | 2nd M1: intention to use \(\int_0^2 xf(x)\,dx\) using their \(f(x)\), must be changed from \(F(x)\); no need for limits |
| \(= \left[\frac{x^3}{9} + \frac{x^2}{12}\right]_0^2\) | A1 | Correct integration (may be unsimplified) |
| \(= \frac{11}{9}\) | A1 | awrt 1.22 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(\text{Var}(X) = \int_0^2 x^2\left(\frac{1}{3}x+\frac{1}{6}\right)dx - \left(\frac{11}{9}\right)^2\) | M1 | 3rd M1: intention to use \(\int x^2 f(x)\,dx - \mu^2\) using their \(f(x)\), must be changed from \(F(x)\); no need for limits |
| \(= \left[\frac{x^4}{12}+\frac{x^3}{18}\right]_0^2 - \left(\frac{11}{9}\right)^2\) | A1ft | 4th A1ft: correct integration; ft their \(E(X)\) |
| \(= \frac{23}{81}\) | A1 | awrt 0.284 |
## Question 3:
### Part (a):
| Working/Answer | Marks | Notes |
|---|---|---|
| $\frac{1}{6}a(a+1) = 0.6$ | M1 | Putting $F(x) = 0.6$ or $1-0.4$ |
| $a^2 + a - 3.6 = 0$ | | |
| $a = \frac{-1 \pm \sqrt{1+4\times3.6}}{2}$ | M1 | Attempting completing the square or quadratic formula (one slip allowed); condone $+$ instead of $\pm$ |
| $= 1.462...$ | | |
| $a = 1.46$ only | A1 | Must reject other root if stated; condone awrt 1.46 |
### Part (b)(i):
| Working/Answer | Marks | Notes |
|---|---|---|
| $f(x) = \frac{d}{dx}F(x) = \frac{1}{3}x + \frac{1}{6}$ | M1A1 | 1st M1: attempting to differentiate $F(x)$, at least one $x^n \to x^{n-1}$ |
| $E(X) = \int_0^2 x\left(\frac{1}{3}x + \frac{1}{6}\right)dx$ | M1 | 2nd M1: intention to use $\int_0^2 xf(x)\,dx$ using their $f(x)$, must be changed from $F(x)$; no need for limits |
| $= \left[\frac{x^3}{9} + \frac{x^2}{12}\right]_0^2$ | A1 | Correct integration (may be unsimplified) |
| $= \frac{11}{9}$ | A1 | awrt 1.22 |
### Part (b)(ii):
| Working/Answer | Marks | Notes |
|---|---|---|
| $\text{Var}(X) = \int_0^2 x^2\left(\frac{1}{3}x+\frac{1}{6}\right)dx - \left(\frac{11}{9}\right)^2$ | M1 | 3rd M1: intention to use $\int x^2 f(x)\,dx - \mu^2$ using their $f(x)$, must be changed from $F(x)$; no need for limits |
| $= \left[\frac{x^4}{12}+\frac{x^3}{18}\right]_0^2 - \left(\frac{11}{9}\right)^2$ | A1ft | 4th A1ft: correct integration; ft their $E(X)$ |
| $= \frac{23}{81}$ | A1 | awrt 0.284 |
---\begin{enumerate}
\item The continuous random variable $X$ has cumulative distribution function given by
\end{enumerate}
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < 0 \\
\frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2 \\
1 & x > 2
\end{array} \right.$$
(a) Find the value of $a$ such that $\mathrm { P } ( X > a ) = 0.4$
Give your answer to 3 significant figures.\\
(b) Use calculus to find (i) $\mathrm { E } ( X )$\\
(ii) $\operatorname { Var } ( X )$.\\
\hfill \mbox{\textit{Edexcel S2 2014 Q3 [11]}}