| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2001 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Expectation of linear transformation |
| Difficulty | Easy -1.2 This is a straightforward S1 question testing basic discrete uniform distribution properties and standard results for linear transformations of random variables. Part (a) requires writing P(Y=y)=1/6, part (b) is recall ('discrete uniform'), and parts (c)-(d) apply the formulas E(aY+b)=aE(Y)+b and Var(aY+b)=a²Var(Y) with E(Y)=3.5 and Var(Y)=35/12 for a fair die. All steps are routine applications of memorized results with no problem-solving required. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(Y=y) = \frac{1}{6}\), \(y = 1,2,3,4,5,6\) | B1, B1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Discrete uniform distribution | B1 (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(Y) = \frac{6+1}{2} = 3.5\) | M1 A1 | |
| \(E(6Y+2) = 6E(Y)+2 = 6\times3.5+2 = 23\) | M1 A1 (4) | Follow through on \(E(Y)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Var(Y) = \frac{7\times5}{12} = \frac{35}{12}\) or \(2.92\) or \(2.916\) | M1 A1 | |
| \(Var(4Y-2) = 16\,Var(Y) = 16\times\frac{35}{12} = \frac{560}{12}\) or \(46.\overline{6}\) or \(46.7\) | M1 M1 A1 (5) | Accept \(46\frac{2}{3}\), \(46.6\), \(46.7\) |
## Question 3:
**Part (a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(Y=y) = \frac{1}{6}$, $y = 1,2,3,4,5,6$ | B1, B1 (2) | |
**Part (b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Discrete uniform distribution | B1 (1) | |
**Part (c)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(Y) = \frac{6+1}{2} = 3.5$ | M1 A1 | |
| $E(6Y+2) = 6E(Y)+2 = 6\times3.5+2 = 23$ | M1 A1 (4) | Follow through on $E(Y)$ |
**Part (d)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Var(Y) = \frac{7\times5}{12} = \frac{35}{12}$ or $2.92$ or $2.916$ | M1 A1 | |
| $Var(4Y-2) = 16\,Var(Y) = 16\times\frac{35}{12} = \frac{560}{12}$ or $46.\overline{6}$ or $46.7$ | M1 M1 A1 (5) | Accept $46\frac{2}{3}$, $46.6$, $46.7$ |
---3. A fair six-sided die is rolled. The random variable $Y$ represents the score on the uppermost, face.
\begin{enumerate}[label=(\alph*)]
\item Write down the probability function of $Y$.
\item State the name of the distribution of $Y$.
Find the value of
\item $\mathrm { E } ( 6 Y + 2 )$,
\item $\operatorname { Var } ( 4 Y - 2 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2001 Q3 [12]}}