| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Standard +0.3 This is a standard S2 probability density function question covering routine techniques: finding k by integration, calculating E(X), finding mode by differentiation, deriving F(x), and evaluating probabilities. All parts follow textbook methods with straightforward polynomial integration. The multi-part structure and 6 marks suggest moderate length, but no step requires novel insight—slightly easier than average A-level. |
| 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 |
|---|---|---|
| \(k\int_1^4(-x^2+5x-4)\,dx = 1\) | M1 | Use of \(\int f(x)\,dx = 1\) |
| \(\therefore k\left[-\frac{x^3}{3}+\frac{5x^2}{2}-4x\right]_1^4 = 1\) | A1 | All correct integration with limits |
| \(\Rightarrow k = \frac{2}{9}\) | A1 | c.s.o.; (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \int_1^4 \frac{2}{9}(-x^3+5x^2-4x)\,dx\) | M1 | Use of \(\int x f(x)\,dx\) |
| \(= \frac{2}{9}\left[-\frac{x^4}{4}+\frac{5x^3}{3}-\frac{4x^2}{2}\right]_1^4\) | A1 | Correct integration with limits |
| \(= \frac{5}{2}\) | A1 | cao; (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d}{dx}f(x) = \frac{2}{9}(-2x+5) = 0 \Rightarrow\) Mode \(= \frac{5}{2}\) | M1, A1 | Differentiate \(f(x)\), set \(= 0\); (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(F(x) = \int_1^{x_0} \frac{2}{9}(-x^2+5x-4)\,dx\) | M1 | Use of \(\int f(x)\,dx\) |
| \(= \left[\frac{2}{9}\left(-\frac{x^3}{3}+\frac{5x^2}{2}-4x\right)\right]_1^{x_0}\) | A1 | Integration with limits |
| \(= \frac{2}{9}\left[-\frac{x^3}{3}+\frac{5x^2}{2}-4x+\frac{11}{6}\right]\) | A1 | a.e.f. |
| \(\therefore F(x) = \begin{cases} 0 & x < 1 \\ \frac{2}{9}\left[-\frac{x^3}{3}+\frac{5x^2}{2}-4x+\frac{11}{6}\right] & 1 \leq x \leq 4 \\ 1 & x > 4 \end{cases}\) | B1, B1 | \(x<1\), \(x>4\); \(1\leq x \leq 4\); (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 2.5) = F(2.5) = 0.5\) | M1, A1 | \(F(2.5)\) or integral; (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Median \(= 2.5\); Distribution is symmetrical | B1, B1 | cao, cao; (2 marks) |
## Question 7:
**(a)**
$k\int_1^4(-x^2+5x-4)\,dx = 1$ | M1 | Use of $\int f(x)\,dx = 1$
$\therefore k\left[-\frac{x^3}{3}+\frac{5x^2}{2}-4x\right]_1^4 = 1$ | A1 | All correct integration with limits
$\Rightarrow k = \frac{2}{9}$ | A1 | c.s.o.; (3 marks)
**(b)**
$E(X) = \int_1^4 \frac{2}{9}(-x^3+5x^2-4x)\,dx$ | M1 | Use of $\int x f(x)\,dx$
$= \frac{2}{9}\left[-\frac{x^4}{4}+\frac{5x^3}{3}-\frac{4x^2}{2}\right]_1^4$ | A1 | Correct integration with limits
$= \frac{5}{2}$ | A1 | cao; (3 marks)
**(c)**
$\frac{d}{dx}f(x) = \frac{2}{9}(-2x+5) = 0 \Rightarrow$ Mode $= \frac{5}{2}$ | M1, A1 | Differentiate $f(x)$, set $= 0$; (2 marks)
[NB: $\frac{5}{2}$ only, no working: B1]
**(d)**
$F(x) = \int_1^{x_0} \frac{2}{9}(-x^2+5x-4)\,dx$ | M1 | Use of $\int f(x)\,dx$
$= \left[\frac{2}{9}\left(-\frac{x^3}{3}+\frac{5x^2}{2}-4x\right)\right]_1^{x_0}$ | A1 | Integration with limits
$= \frac{2}{9}\left[-\frac{x^3}{3}+\frac{5x^2}{2}-4x+\frac{11}{6}\right]$ | A1 | a.e.f.
$\therefore F(x) = \begin{cases} 0 & x < 1 \\ \frac{2}{9}\left[-\frac{x^3}{3}+\frac{5x^2}{2}-4x+\frac{11}{6}\right] & 1 \leq x \leq 4 \\ 1 & x > 4 \end{cases}$ | B1, B1 | $x<1$, $x>4$; $1\leq x \leq 4$; (5 marks)
**(e)**
$P(X \leq 2.5) = F(2.5) = 0.5$ | M1, A1 | $F(2.5)$ or integral; (2 marks)
**(f)**
Median $= 2.5$; Distribution is symmetrical | B1, B1 | cao, cao; (2 marks)7. The random variable $X$ has probability density function
$$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4 \\ 0 , & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { 2 } { 9 }$.
Find
\item $\mathrm { E } ( X )$,
\item the mode of $X$.
\item the cumulative distribution function $\mathrm { F } ( x )$ for all $x$.
\item Evaluate $\mathrm { P } ( X \leq 2.5 )$,
\item Deduce the value of the median and comment on the shape of the distribution.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2005 Q7 [17]}}