| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Standard +0.3 This is a straightforward application of standard uniform distribution formulas. Part (a) requires recalling E(X) = (a+b)/2 and Var(X) = (b-a)²/12, then solving two simultaneous equations. Part (b) uses basic probability properties with the uniform distribution. All steps are routine with no conceptual challenges beyond formula recall and algebraic manipulation. |
| Spec | 5.02e Discrete uniform distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}(a+b) = 23\) and \(\frac{1}{12}(b-a)^2 = 75\) | B1 B1 | 1st B1: at least one correct equation; 2nd B1: any 2 correct equations using both 23 and 75 |
| \(a+b = 46\) and \(b-a = \sqrt{12 \times 75} (=30)\) | M1 | For rearranging to get two linear equations in \(a\) and \(b\), or rearranging and substituting linear equation into quadratic |
| Adding gives \(2b = 76\) | M1 | For solving i.e. eliminating one variable leading to linear equation, or solving quadratic correctly |
| \(b = 38\) and \(a = 8\) | A1 A1 | 1st A1: \(b=38\); 2nd A1: \(a=8\). SC: if \(b=8\), \(a=38\) or two sets given, max B1B1M1M1A1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a+b=46\) hence \((46-2a)^2 = 900\) | M1 | |
| \(a^2 - 46a + 304 = 0\) | ||
| \((a-8)(a-38)=0\) | M1 | |
| \(b=38\) and \(a=8\) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(23 < X < c) = 0.5 - 0.32\) or \(c = 28.4\) and \(\text{prob} = \frac{5.4}{30}\) | M1 | For a correct method, e.g. correct expression or seeing calculation for \(c\) and calculation for probability |
| \(= 0.18\) | A1 | For 0.18 only |
## Question 3:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}(a+b) = 23$ and $\frac{1}{12}(b-a)^2 = 75$ | B1 B1 | 1st B1: at least one correct equation; 2nd B1: any 2 correct equations using both 23 and 75 |
| $a+b = 46$ and $b-a = \sqrt{12 \times 75} (=30)$ | M1 | For rearranging to get two linear equations in $a$ and $b$, or rearranging and substituting linear equation into quadratic |
| Adding gives $2b = 76$ | M1 | For solving i.e. eliminating one variable leading to linear equation, or solving quadratic correctly |
| $b = 38$ and $a = 8$ | A1 A1 | 1st A1: $b=38$; 2nd A1: $a=8$. SC: if $b=8$, $a=38$ or two sets given, max B1B1M1M1A1A0 |
**Alternative:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a+b=46$ hence $(46-2a)^2 = 900$ | M1 | |
| $a^2 - 46a + 304 = 0$ | | |
| $(a-8)(a-38)=0$ | M1 | |
| $b=38$ and $a=8$ | A1 A1 | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(23 < X < c) = 0.5 - 0.32$ or $c = 28.4$ and $\text{prob} = \frac{5.4}{30}$ | M1 | For a correct method, e.g. correct expression or seeing calculation for $c$ and calculation for probability |
| $= 0.18$ | A1 | For 0.18 only |
---3. The random variable $X$ has a continuous uniform distribution on $[ a , b ]$ where $a$ and $b$ are positive numbers.
Given that $\mathrm { E } ( X ) = 23$ and $\operatorname { Var } ( X ) = 75$
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$.
Given that $\mathrm { P } ( X > c ) = 0.32$
\item find $\mathrm { P } ( 23 < X < c )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2013 Q3 [8]}}