| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard uniform distribution formulas. Part (a) requires solving two simultaneous equations using E(X) = (α+β)/2 and the probability formula. Parts (b) and (c) apply basic probability calculations. While multi-part, each step uses direct formula application with minimal problem-solving insight required—slightly easier than average A-level. |
| Spec | 5.02e Discrete uniform distribution5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(X) = \frac{\alpha + \beta}{2} = 3.5 \Rightarrow \alpha + \beta = 7\) | B1 | Correct equation; need not be simplified |
| \(P(X > 5) = \frac{\beta - 5}{\beta - \alpha} = \frac{2}{5} \Rightarrow 5(\beta - 5) = 2(\beta - \alpha)\) | M1 | A second correct equation; using simultaneous equations and eliminating \(\alpha\) or \(\beta\) to gain a value of \(\alpha\) and \(\beta\) |
| \(\alpha = -4\) | A1 | 1st A1 for \(-4\) |
| \(\beta = 11\) | A1 | 2nd A1 for \(11\); NB award full marks for \(\alpha = -4, \beta = 11\) (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{c+4}{15} = \frac{2}{3}\) | ||
| \([c =]\ 6\) | B1 | B1 for 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(6 < X < 9) = \frac{1}{15} \times (3)\) | M1 | \(\frac{1}{\beta - \alpha} \times (9 - c)\) or \([F(9) - F(c)] = \frac{13}{15} - \frac{2}{3}\); SC if \(9 >\) "their \(b\)" award for \(1 - \frac{2}{3}\) |
| \(= 0.2\) | A1cso | 0.2 oe (3 marks total for b) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([P(S < 45)] = \frac{3}{10}\) | B1 | \(\frac{3}{10}\) seen; does not need to be associated with \(P(S < 45)\) |
| \([P(S > 55)] = \frac{1}{2}\) | B1 | \(\frac{1}{2}\) seen; does not need to be associated with \(P(S > 55)\) |
| total \(= \frac{3}{10} + \frac{1}{2} = \frac{4}{5}\) | M1A1 | M1 for adding their two areas with total \(< 1\); do not allow \(2'\) a single area; A1 \(\frac{4}{5}\) oe; NB award full marks for \(\frac{4}{5}\) (4 marks) |
## Question 4:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(X) = \frac{\alpha + \beta}{2} = 3.5 \Rightarrow \alpha + \beta = 7$ | B1 | Correct equation; need not be simplified |
| $P(X > 5) = \frac{\beta - 5}{\beta - \alpha} = \frac{2}{5} \Rightarrow 5(\beta - 5) = 2(\beta - \alpha)$ | M1 | A second correct equation; using simultaneous equations and eliminating $\alpha$ or $\beta$ to gain a value of $\alpha$ and $\beta$ |
| $\alpha = -4$ | A1 | 1st A1 for $-4$ |
| $\beta = 11$ | A1 | 2nd A1 for $11$; **NB** award full marks for $\alpha = -4, \beta = 11$ **(4 marks)** |
### Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{c+4}{15} = \frac{2}{3}$ | | |
| $[c =]\ 6$ | B1 | B1 for 6 |
### Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(6 < X < 9) = \frac{1}{15} \times (3)$ | M1 | $\frac{1}{\beta - \alpha} \times (9 - c)$ or $[F(9) - F(c)] = \frac{13}{15} - \frac{2}{3}$; **SC** if $9 >$ "their $b$" award for $1 - \frac{2}{3}$ |
| $= 0.2$ | A1cso | 0.2 oe **(3 marks total for b)** |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[P(S < 45)] = \frac{3}{10}$ | B1 | $\frac{3}{10}$ seen; does not need to be associated with $P(S < 45)$ |
| $[P(S > 55)] = \frac{1}{2}$ | B1 | $\frac{1}{2}$ seen; does not need to be associated with $P(S > 55)$ |
| total $= \frac{3}{10} + \frac{1}{2} = \frac{4}{5}$ | M1A1 | M1 for adding their two areas with total $< 1$; do not allow $2'$ a single area; **A1** $\frac{4}{5}$ oe; **NB** award full marks for $\frac{4}{5}$ **(4 marks)** |
---4. The continuous random variable $X$ is uniformly distributed over the interval $[ \alpha , \beta ]$
Given that $\mathrm { E } ( X ) = 3.5$ and $\mathrm { P } ( X > 5 ) = \frac { 2 } { 5 }$
\begin{enumerate}[label=(\alph*)]
\item find the value of $\alpha$ and the value of $\beta$
Given that $\mathrm { P } ( X < c ) = \frac { 2 } { 3 }$
\item \begin{enumerate}[label=(\roman*)]
\item find the value of $c$
\item find $\mathrm { P } ( c < X < 9 )$
A rectangle has a perimeter of 200 cm . The length, $S \mathrm {~cm}$, of one side of this rectangle is uniformly distributed between 30 cm and 80 cm .
\end{enumerate}\item Find the probability that the length of the shorter side of the rectangle is less than 45 cm .
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2017 Q4 [11]}}