| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate and compare mean, median, mode |
| Difficulty | Moderate -0.3 This is a standard S2 probability density function question covering routine techniques: integrating to find k, deriving the CDF, calculating expectation, and finding median/mode. All parts follow textbook methods with straightforward integration of linear functions. The skewness part requires only comparing mean/median/mode positions, not calculation. Slightly easier than average due to the simple linear pdf form. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\int_1^{4} \frac{1 + x}{k} dx = 1\) | \(\int f(x) = 1\) Area = 1 | M1 |
| \(\therefore \left[\frac{x}{k} + \frac{x^2}{2k}\right]_1^4 = 1\) | correct integral/correct expression | A1 |
| \(k = \frac{21}{2}\) ✱ | cso | A1 |
| (b) \(P(X \leq x_0) = \int_1^{x_0} \frac{2}{21}(1 + x)\) | \(\int f(x)\) variable limit or +C | M1 |
| \(= \left[\frac{2x}{21} + \frac{x^2}{21}\right]_{1}^{x_0}\) | correct integral + limit of 1 | A1 |
| \(= \frac{2x_0 + x_0^2 - 3}{21}\) or \(\frac{(3+x)(x-1)}{21}\) | A1 | |
| \[F(x) = \begin{cases} 0, & x < 1 \\ \frac{x^2 + 2x - 3}{21}, & 1 \leq x < 4 \\ 1, & x \geq 4 \end{cases}\] | middle; ends | B1√; B1 |
| (c) \(E(X) = \int_1^4 \frac{2x}{21}(1 + x) dx\) | valid attempt \(\int xf(x)\) | M1 |
| \(x^2\) and \(x^3\) | A1 | |
| \(= \left[\frac{x^2}{21} + \frac{2x^3}{63}\right]_1^4\) | correct integration | A1 |
| \(= \frac{171}{63} = 2\frac{5}{7} = \frac{19}{7} = 2.7142 \ldots\) | awrt 2.71 | A1 |
| (d) \(F(m) = 0.5 \Rightarrow \frac{x^2 + 2x - 3}{21} = \frac{1}{2}\) | putting their \(F(x) = 0.5\) | M1 |
| \(\therefore 2x^2 + 4x - 27 = 0\) or equiv | attempt their 3 term quadratic | M1 |
| \(\therefore x = \frac{-4 \pm \sqrt{16 - 4.2(-27)}}{4}\) | A1 | |
| \(\therefore x = -1 \pm 3.8078 \ldots\) | awrt 2.81 | A1 |
| i.e. \(x = 2.8078 \ldots\) | ||
| (e) Mode = 4 | B1 | |
| (f) Mean < median < mode (\(\Rightarrow\) negative skew) OR Mean < median | allow numbers in place of words | B1; B1 |
| w diagram but line must not cross y axis | [diagram shown with horizontal segment on x-axis from origin, then increasing curve] |
**(a)** $\int_1^{4} \frac{1 + x}{k} dx = 1$ | $\int f(x) = 1$ Area = 1 | M1
$\therefore \left[\frac{x}{k} + \frac{x^2}{2k}\right]_1^4 = 1$ | correct integral/correct expression | A1
$k = \frac{21}{2}$ ✱ | cso | A1 |
**(b)** $P(X \leq x_0) = \int_1^{x_0} \frac{2}{21}(1 + x)$ | $\int f(x)$ variable limit or +C | M1
$= \left[\frac{2x}{21} + \frac{x^2}{21}\right]_{1}^{x_0}$ | correct integral + limit of 1 | A1
$= \frac{2x_0 + x_0^2 - 3}{21}$ or $\frac{(3+x)(x-1)}{21}$ | A1 |
$$F(x) = \begin{cases} 0, & x < 1 \\ \frac{x^2 + 2x - 3}{21}, & 1 \leq x < 4 \\ 1, & x \geq 4 \end{cases}$$ | middle; ends | B1√; B1 |
**(c)** $E(X) = \int_1^4 \frac{2x}{21}(1 + x) dx$ | valid attempt $\int xf(x)$ | M1
$x^2$ and $x^3$ | A1 |
$= \left[\frac{x^2}{21} + \frac{2x^3}{63}\right]_1^4$ | correct integration | A1 |
$= \frac{171}{63} = 2\frac{5}{7} = \frac{19}{7} = 2.7142 \ldots$ | awrt 2.71 | A1 |
**(d)** $F(m) = 0.5 \Rightarrow \frac{x^2 + 2x - 3}{21} = \frac{1}{2}$ | putting their $F(x) = 0.5$ | M1
$\therefore 2x^2 + 4x - 27 = 0$ or equiv | attempt their 3 term quadratic | M1
$\therefore x = \frac{-4 \pm \sqrt{16 - 4.2(-27)}}{4}$ | A1 |
$\therefore x = -1 \pm 3.8078 \ldots$ | awrt 2.81 | A1 |
i.e. $x = 2.8078 \ldots$ |
**(e)** Mode = 4 | B1 |
**(f)** Mean < median < mode ($\Rightarrow$ negative skew) OR Mean < median | allow numbers in place of words | B1; B1 |
w diagram but line must not cross y axis | [diagram shown with horizontal segment on x-axis from origin, then increasing curve] |The continuous random variable $X$ has probability density function
$$f(x) = \begin{cases}
\frac{1+x}{k}, & 1 \leqslant x \leqslant 4, \\
0, & \text{otherwise}.
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac{21}{2}$. [3]
\item Specify fully the cumulative distribution function of $X$. [5]
\item Calculate E$(X)$. [3]
\item Find the value of the median. [3]
\item Write down the mode. [1]
\item Explain why the distribution is negatively skewed. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q6 [16]}}