| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Variance of linear transformation |
| Difficulty | Easy -1.2 This is a straightforward application of standard results for expectation and variance of linear transformations. Part (a) requires recall of the discrete uniform distribution formula or simple calculation; parts (b) and (c) are direct applications of E(aX+b) and Var(aX+b) rules with no problem-solving required. This is easier than average A-level content. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.02e Discrete uniform distribution |
| Answer | Marks |
|---|---|
| \(E(X) = 3\) | B1 |
| \(\text{Var}(X) = \frac{25-1}{12} = 2\) | M1A1 |
| AG | |
| \(\text{Var}(X) = 1^2 \times \frac{1}{5} + 2^2 \times \frac{1}{5} + 3^2 \times \frac{1}{5} + \ldots - 3^2 = 11 - 9 = 2\) | |
| AG | |
| Accept (55/5)-9 as minimum evidence. |
| Answer | Marks |
|---|---|
| \(E(3X - 2) = 3E(X) - 2 = 7\) | M1A1 |
| Score 2 marks |
| Answer | Marks |
|---|---|
| \(\text{Var}(4 - 3x) = 3^2 \text{Var}(X) = 18\) | M1A1 |
| Score 2 marks |
**Part (a):**
| $E(X) = 3$ | B1 |
| $\text{Var}(X) = \frac{25-1}{12} = 2$ | M1A1 |
| **AG** | |
| $\text{Var}(X) = 1^2 \times \frac{1}{5} + 2^2 \times \frac{1}{5} + 3^2 \times \frac{1}{5} + \ldots - 3^2 = 11 - 9 = 2$ | |
| **AG** | |
| Accept (55/5)-9 as minimum evidence. | |
**Part (b):**
| $E(3X - 2) = 3E(X) - 2 = 7$ | M1A1 |
| **Score 2 marks** | |
**Part (c):**
| $\text{Var}(4 - 3x) = 3^2 \text{Var}(X) = 18$ | M1A1 |
| **Score 2 marks** | |
**Total 7 marks**
---\begin{enumerate}
\item The random variable $X$ has the discrete uniform distribution
\end{enumerate}
$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } , \quad x = 1,2,3,4,5$$
(a) Write down the value of $\mathrm { E } ( X )$ and show that $\operatorname { Var } ( X ) = 2$.
Find\\
(b) $\mathrm { E } ( 3 X - 2 )$,\\
(c) $\operatorname { Var } ( 4 - 3 X )$.\\
\hfill \mbox{\textit{Edexcel S1 2006 Q4 [7]}}