| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a standard S1 probability distribution question requiring routine application of formulas. Part (a) uses basic probability axioms (sum=1) and expectation definition to form simultaneous equations. Parts (b)-(f) involve straightforward substitution and application of standard variance/expectation formulas. While multi-part with 6 sections, each step is mechanical with no problem-solving insight required—slightly easier than average due to its highly procedural nature. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| \(x\) | 1 | 3 | 5 | 7 | 9 |
| \(\mathrm { P } ( X = x )\) | 0.2 | \(p\) | 0.2 | \(q\) | 0.15 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(p + q = 0.45\) | B1 | \(0.55 + p + q = 1\) award B1; not seen award B0 |
| \(\sum xP(X=x) = 4.5\) | M1 | \(0.2+3p+1+7q+1.35=4.5\) or equivalent |
| \(3p + 7q = 1.95\) | A1 | \(3p+7q+k=4.5\) award M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Attempt to solve equations in (a) | M1 | Must involve 2 linear equations in 2 unknowns |
| \(q = 0.15\) | A1 | Correct answers only for accuracy |
| \(p = 0.30\) | A1 | Correct answers with no working award 3/3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(4 < X < 7) = P(5) + P(7)\) | M1 | |
| \(= 0.2 + q = 0.35\) | A1\(\int\) | Follow through accuracy mark for their \(q\), \(0 < q < 0.8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\text{Var}(X) = E(X^2) - [E(X)]^2 = 27.4 - 4.5^2\) | M1 | Attempt to substitute given values only into correct formula |
| \(= 7.15\) | A1 | \(7.15\) seen award 2/2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(19-4X) = 19 - 4 \times 4.5 = 1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\text{Var}(19-4X) = 16\text{Var}(X)\) | M1 | Accept 'invisible brackets' i.e. \(-4^2\ \text{Var}(X)\) provided answer positive |
| \(= 16 \times 7.15 = 114.4\) | A1 | Anything that rounds to \(114\) |
## Question 7:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $p + q = 0.45$ | B1 | $0.55 + p + q = 1$ award B1; not seen award B0 |
| $\sum xP(X=x) = 4.5$ | M1 | $0.2+3p+1+7q+1.35=4.5$ or equivalent |
| $3p + 7q = 1.95$ | A1 | $3p+7q+k=4.5$ award M1 |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Attempt to solve equations in (a) | M1 | Must involve 2 linear equations in 2 unknowns |
| $q = 0.15$ | A1 | Correct answers only for accuracy |
| $p = 0.30$ | A1 | Correct answers with no working award 3/3 |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(4 < X < 7) = P(5) + P(7)$ | M1 | |
| $= 0.2 + q = 0.35$ | A1$\int$ | Follow through accuracy mark for their $q$, $0 < q < 0.8$ |
### Part (d):
| Working | Mark | Guidance |
|---------|------|----------|
| $\text{Var}(X) = E(X^2) - [E(X)]^2 = 27.4 - 4.5^2$ | M1 | Attempt to substitute given values only into correct formula |
| $= 7.15$ | A1 | $7.15$ seen award 2/2 |
### Part (e):
| Working | Mark | Guidance |
|---------|------|----------|
| $E(19-4X) = 19 - 4 \times 4.5 = 1$ | B1 | |
### Part (f):
| Working | Mark | Guidance |
|---------|------|----------|
| $\text{Var}(19-4X) = 16\text{Var}(X)$ | M1 | Accept 'invisible brackets' i.e. $-4^2\ \text{Var}(X)$ provided answer positive |
| $= 16 \times 7.15 = 114.4$ | A1 | Anything that rounds to $114$ |7. The random variable $X$ has probability distribution
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 1 & 3 & 5 & 7 & 9 \\
\hline
$\mathrm { P } ( X = x )$ & 0.2 & $p$ & 0.2 & $q$ & 0.15 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { E } ( X ) = 4.5$, write down two equations involving $p$ and $q$.
Find
\item the value of $p$ and the value of $q$,
\item $\mathrm { P } ( 4 < X \leqslant 7 )$.
Given that $\mathrm { E } \left( X ^ { 2 } \right) = 27.4$, find
\item $\operatorname { Var } ( X )$,
\item $\mathrm { E } ( 19 - 4 X )$,
\item $\operatorname { Var } ( 19 - 4 X )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2007 Q7 [13]}}