| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Probability distribution from formula |
| Difficulty | Moderate -0.3 This is a straightforward S1 probability distribution question requiring systematic application of standard formulas. Part (a) uses ΣP(X=x)=1 to find k, parts (b-d) involve direct calculation of probabilities and expectations, and part (e) applies the variance formula Var(aX+b)=a²Var(X). While multi-part with several calculations, each step follows routine procedures with no conceptual challenges or problem-solving insight required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2k+k+k+5k=1\) | M1 | A correct expression using \(\sum p(x)=1\) |
| \(k=\dfrac{1}{9}\) | A1cso | Given answer with no incorrect working seen. \(9k=1\) with no working scores M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(1\leqslant X < 4)=\dfrac{2}{9}\) | B1 | Allow recurring decimals; ISW after correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X)=(1x)\frac{1}{9}+2\times\frac{1}{9}+4\times\frac{5}{9}\) or \(E(X)=(1x)k+2\times k+4\times5k\) | M1 | Use of \(\sum xp(x)\); 3 non-zero terms, at most 1 error or omission |
| \(=\dfrac{23}{9}\) | A1 | Allow exact equivalent e.g. \(2\frac{5}{9}\) or \(2.\dot{5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X^2)=1\times\frac{1}{9}+2^2\times\frac{1}{9}+4^2\times\frac{5}{9}\) | M1 | Use of \(\sum x^2p(x)\); 3 non-zero terms, at most 1 error or omission |
| \(=\dfrac{85}{9}\) | A1 | Allow exact equivalent e.g. \(9\frac{4}{9}\) or \(9.\dot{4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(X)=\frac{85}{9}-\left(\frac{23}{9}\right)^2\) | M1 | Use of \(E(X^2)-[E(X)]^2\) |
| \(\text{Var}(3X+1)=9\times\left(\frac{85}{9}-\left(\frac{23}{9}\right)^2\right)\) | M1 | Writing or using \(9\times\text{Var}(X)\); \(\left[9\times\frac{85}{9}\text{ on its own is M0}\right]\) |
| \(=\dfrac{236}{9}\) | A1 | Allow exact equivalent e.g. \(26\frac{2}{9}\) or \(26.\dot{2}\) |
# Question 4:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2k+k+k+5k=1$ | M1 | A correct expression using $\sum p(x)=1$ |
| $k=\dfrac{1}{9}$ | A1cso | Given answer with no incorrect working seen. $9k=1$ with no working scores M0A0 |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(1\leqslant X < 4)=\dfrac{2}{9}$ | B1 | Allow recurring decimals; ISW after correct answer |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X)=(1x)\frac{1}{9}+2\times\frac{1}{9}+4\times\frac{5}{9}$ or $E(X)=(1x)k+2\times k+4\times5k$ | M1 | Use of $\sum xp(x)$; 3 non-zero terms, at most 1 error or omission |
| $=\dfrac{23}{9}$ | A1 | Allow exact equivalent e.g. $2\frac{5}{9}$ or $2.\dot{5}$ |
## Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X^2)=1\times\frac{1}{9}+2^2\times\frac{1}{9}+4^2\times\frac{5}{9}$ | M1 | Use of $\sum x^2p(x)$; 3 non-zero terms, at most 1 error or omission |
| $=\dfrac{85}{9}$ | A1 | Allow exact equivalent e.g. $9\frac{4}{9}$ or $9.\dot{4}$ |
## Part (e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(X)=\frac{85}{9}-\left(\frac{23}{9}\right)^2$ | M1 | Use of $E(X^2)-[E(X)]^2$ |
| $\text{Var}(3X+1)=9\times\left(\frac{85}{9}-\left(\frac{23}{9}\right)^2\right)$ | M1 | Writing or using $9\times\text{Var}(X)$; $\left[9\times\frac{85}{9}\text{ on its own is M0}\right]$ |
| $=\dfrac{236}{9}$ | A1 | Allow exact equivalent e.g. $26\frac{2}{9}$ or $26.\dot{2}$ |
---4. A discrete random variable $X$ has probability function
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l }
k ( 2 - x ) & x = 0,1 \\
k ( 3 - x ) & x = 2,3 \\
k ( x + 1 ) & x = 4 \\
0 & \text { otherwise }
\end{array} \right.$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { 1 } { 9 }$
Find the exact value of
\item $\mathrm { P } ( 1 \leqslant X < 4 )$
\item $\mathrm { E } ( X )$
\item $\mathrm { E } \left( X ^ { 2 } \right)$
\item $\operatorname { Var } ( 3 X + 1 )$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2018 Q4 [10]}}