| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Name the distribution |
| Difficulty | Easy -1.2 This is a straightforward multi-part question testing basic knowledge of discrete uniform distribution. Part (a) requires simple recognition, parts (b)-(d) involve direct probability calculations, and parts (e)-(g) test standard expectation and variance formulas. Part (h) requires using the given information to find 'a' then apply variance properties. All parts are routine applications of S1 content with no problem-solving or novel insight required. |
| Spec | 5.02e Discrete uniform distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | (Discrete) Uniform | B1 |
| (b) | \(P(X = 4) = \frac{1}{5}\) oe | B1 |
| (c) | \(F(3) = \frac{3}{5}\) oe | B1 |
| (d) | \(P(3X - 3 > X + 4) = P(X > 3.5)\) | M1 |
| \(= \frac{2}{5}\) oe | A1 | A1 for correct answer |
| (e) | \(E(X) = 3\) | B1 |
| (f) | \(E(X^2) = \frac{1}{5}\left(1^2 + 2^2 + 3^2 + 4^2 + 5^2\right)\) | M1 |
| \(= 11\) | A1 | A1 for 11 |
| (g) | \(\text{Var}(X) = 11 - 3^2\) or \(\frac{(5+1)(5-1)}{12}\) | M1 |
| \(= 2\) | A1 | A1 for 2 |
| (h) | \(11.4 = aE(X) - 3\) or \(11.4 = 3a - 3\) | M1 |
| \(a = 4.8\) | A1 | |
| \(\text{Var}(4.8X - 3) = '4.8'^2 \times '2'\) | M1 | 2nd M1 for 'their \(a^2\) '×' their Var(X)' (must see values substituted) (may be implied by correct answer or correct ft answer). NB 'their Var(X)' < 0 is M0 here. |
| \(= 46.08\) | A1 | A1 awrt 46.1 |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | (Discrete) Uniform | B1 | B1 for uniform |
| (b) | $P(X = 4) = \frac{1}{5}$ oe | B1 | |
| (c) | $F(3) = \frac{3}{5}$ oe | B1 | |
| (d) | $P(3X - 3 > X + 4) = P(X > 3.5)$ | M1 | M1 for identifying correct probabilities i.e. $P(X > 3.5)$ or $P(X = 4) + P(X = 5)$ |
| | $= \frac{2}{5}$ oe | A1 | A1 for correct answer |
| (e) | $E(X) = 3$ | B1 | |
| (f) | $E(X^2) = \frac{1}{5}\left(1^2 + 2^2 + 3^2 + 4^2 + 5^2\right)$ | M1 | M1 for correct expression |
| | $= 11$ | A1 | A1 for 11 |
| (g) | $\text{Var}(X) = 11 - 3^2$ or $\frac{(5+1)(5-1)}{12}$ | M1 | M1 for either 'their (f)' − 'their (e)²' or correct expression $\frac{(5+1)(5-1)}{12}$ |
| | $= 2$ | A1 | A1 for 2 |
| (h) | $11.4 = aE(X) - 3$ or $11.4 = 3a - 3$ | M1 | 1st M1 for setting up correct linear equation using $aE(X) - 3 = 11.4$. 1st A1 may be implied by correct answer |
| | $a = 4.8$ | A1 | |
| | $\text{Var}(4.8X - 3) = '4.8'^2 \times '2'$ | M1 | 2nd M1 for 'their $a^2$ '×' their Var(X)' (must see values substituted) (may be implied by correct answer or correct ft answer). NB 'their Var(X)' < 0 is M0 here. |
| | $= 46.08$ | A1 | A1 awrt 46.1 |
**Total: 14 marks**
---3. The discrete random variable $X$ has probability distribution
$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$
\begin{enumerate}[label=(\alph*)]
\item Write down the name given to this distribution.
Find
\item $\mathrm { P } ( X = 4 )$
\item $\mathrm { F } ( 3 )$
\item $\mathrm { P } ( 3 X - 3 > X + 4 )$
\item Write down $\mathrm { E } ( X )$
\item Find $\mathrm { E } \left( X ^ { 2 } \right)$
\item Hence find $\operatorname { Var } ( X )$
Given that $\mathrm { E } ( a X - 3 ) = 11.4$
\item find $\operatorname { Var } ( a X - 3 )$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2016 Q3 [14]}}