| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Continuous CDF with polynomial pieces |
| Difficulty | Standard +0.3 This is a standard S2 CDF question requiring continuity conditions to find constants, differentiation to find the pdf, and integration for expectation. Part (d) involves solving a cubic equation numerically. All techniques are routine for S2 with no novel insight 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(F(1) = 0\): \(\frac{4}{10} + a + b = 0\) | M1 | Using \(F(1) = 0\); clear attempt to form linear equation in \(a\) and \(b\) |
| \(a = -\frac{3}{5}\) or \(b = \frac{1}{5}\) | A1 | Either value correct |
| \(F(2) = 1\): \(2 + 2a + b = 1\) | M1 | Using \(F(2) = 1\); clear attempt to form second linear equation |
| \(a = -\frac{3}{5},\ b = \frac{1}{5}\) | A1 | Both correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Differentiating cdf: \(f(x) = \frac{3}{10}x^2 + \frac{6}{10}x + a,\quad 1 \leq x \leq 2\) | B1 cso | Must differentiate then factorise |
| \(= \frac{3}{10}(x^2 + 2x - 2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(X) = \int_1^2 \frac{3}{10}(x^3 + 2x^2 - 2x)\,dx\) | M1 | Clear attempt to use \(xf(x)\) with intention to integrate |
| \(= \frac{3}{10}\left[\frac{1}{4}x^4 + \frac{2}{3}x^3 - x^2\right]_1^2\) | M1d A1 | Dependent on previous M; at least one correct term with correct coefficient |
| \(= \frac{13}{8}\) | A1 | Accept 1.63 and 1.625 or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(F(1.425) = 0.24355\), \(F(1.435) = 0.25227\) | M1A1 | Expression showing substitution of 1.425 or 1.435 into \(F(x)\); awrt 0.244 and 0.252 |
| \(0.25\) lies between \(F(1.425)\) and \(F(1.435)\), hence result | A1 | Statement must be true for their method |
# Question 5:
## Part 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $F(1) = 0$: $\frac{4}{10} + a + b = 0$ | M1 | Using $F(1) = 0$; clear attempt to form linear equation in $a$ and $b$ |
| $a = -\frac{3}{5}$ or $b = \frac{1}{5}$ | A1 | Either value correct |
| $F(2) = 1$: $2 + 2a + b = 1$ | M1 | Using $F(2) = 1$; clear attempt to form second linear equation |
| $a = -\frac{3}{5},\ b = \frac{1}{5}$ | A1 | Both correct |
## Part 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Differentiating cdf: $f(x) = \frac{3}{10}x^2 + \frac{6}{10}x + a,\quad 1 \leq x \leq 2$ | B1 cso | Must differentiate then factorise |
| $= \frac{3}{10}(x^2 + 2x - 2)$ | | |
## Part 5(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = \int_1^2 \frac{3}{10}(x^3 + 2x^2 - 2x)\,dx$ | M1 | Clear attempt to use $xf(x)$ with intention to integrate |
| $= \frac{3}{10}\left[\frac{1}{4}x^4 + \frac{2}{3}x^3 - x^2\right]_1^2$ | M1d A1 | Dependent on previous M; at least one correct term with correct coefficient |
| $= \frac{13}{8}$ | A1 | Accept 1.63 and 1.625 or exact equivalent |
## Part 5(d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $F(1.425) = 0.24355$, $F(1.435) = 0.25227$ | M1A1 | Expression showing substitution of 1.425 or 1.435 into $F(x)$; awrt 0.244 and 0.252 |
| $0.25$ lies between $F(1.425)$ and $F(1.435)$, hence result | A1 | Statement must be true for their method |
---\begin{enumerate}
\item The continuous random variable $X$ has a cumulative distribution function
\end{enumerate}
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r }
0 & x < 1 \\
\frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b & 1 \leqslant x \leqslant 2 \\
1 & x > 2
\end{array} \right.$$
where $a$ and $b$ are constants.\\
(a) Find the value of $a$ and the value of $b$.\\
(b) Show that $\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leqslant x \leqslant 2$\\
(c) Use integration to find $\mathrm { E } ( X )$.\\
(d) Show that the lower quartile of $X$ lies between 1.425 and 1.435
\hfill \mbox{\textit{Edexcel S2 2013 Q5 [12]}}