| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | January |
| Marks | 11 |
| 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 textbook exercise requiring routine application of probability axioms (sum=1) and expectation formula to find two unknowns, followed by straightforward calculations of CDF, probability, and variance using standard formulas. All steps are mechanical with no problem-solving insight required, making it slightly easier than average but not trivial due to the multi-part nature and algebraic manipulation needed. |
| Spec | 2.02f Measures of average and spread2.04a Discrete probability distributions |
| \(x\) | - 1 | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(\frac { 1 } { 10 }\) | \(\frac { 1 } { 5 }\) | \(\frac { 3 } { 10 }\) | \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a + \frac{1}{10} + \frac{1}{5} + \frac{3}{10} + b = 1\) or \(a + b = \frac{2}{5}\) | M1 | Correct linear equation in \(a\) and \(b\) based on sum of probs. \(= 1\) |
| \(-a + 0 + \frac{1}{5} + \frac{6}{10} + 3b = \frac{9}{5}\) or \(3b - a = 1\) | M1 | Attempt at second linear equation based on \(E(X) = 1.8\); allow one slip |
| \(a = \frac{1}{20},\ b = \frac{7}{20}\) | M1A1 | M1 for attempt to solve; A1 for \(a = 0.05\) and \(b = 0.35\) or exact fraction equivalents |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(F(x)\) table completed correctly with at least 4 correct values | B1 | Condone no \(F(x)\) or \(P(x)\) if in correct table |
| All values of \(F(x)\) correct: \(0.05, 0.15, 0.35, 0.65, 1\) | B1 | \(F(-1)=0.05\) etc must be stated if not in table |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X < 2.5) = F(2) = \frac{13}{20}\) or \(0.65\) | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X^2) = 1\times0.05 + 0 + 1\times0.2 + 4\times0.3 + 9\times0.35 = 4.6\) or \(\frac{23}{5}\) | M1 | At least 3 correct terms or sight of 4.6 or correct \(\text{Var}(X)\); NB \(\frac{4.6}{5}\) loses M1 |
| \(\text{Var}(X) = E(X^2) - 1.8^2 = 1.36\) or \(\frac{34}{25}\) | M1 | Follow through their "4.6"; must see \(-1.8^2\) or 1.36 |
| \(\text{Var}(3-2X) = (-2)^2 \text{Var}(X)\) | M1 | Correct use of \(\text{Var}(aX+b)\) formula; condone \(-2^2\) if this later becomes \(+4\) |
| \(= 4 \times 1.36 = 5.44\) | A1 | Accept \(\frac{136}{25}\) or exact equivalent |
# Question 4:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a + \frac{1}{10} + \frac{1}{5} + \frac{3}{10} + b = 1$ or $a + b = \frac{2}{5}$ | M1 | Correct linear equation in $a$ and $b$ based on sum of probs. $= 1$ |
| $-a + 0 + \frac{1}{5} + \frac{6}{10} + 3b = \frac{9}{5}$ or $3b - a = 1$ | M1 | Attempt at second linear equation based on $E(X) = 1.8$; allow one slip |
| $a = \frac{1}{20},\ b = \frac{7}{20}$ | M1A1 | M1 for attempt to solve; A1 for $a = 0.05$ and $b = 0.35$ or exact fraction equivalents |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $F(x)$ table completed correctly with at least 4 correct values | B1 | Condone no $F(x)$ or $P(x)$ if in correct table |
| All values of $F(x)$ correct: $0.05, 0.15, 0.35, 0.65, 1$ | B1 | $F(-1)=0.05$ etc must be stated if not in table |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X < 2.5) = F(2) = \frac{13}{20}$ or $0.65$ | B1 | cao |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X^2) = 1\times0.05 + 0 + 1\times0.2 + 4\times0.3 + 9\times0.35 = 4.6$ or $\frac{23}{5}$ | M1 | At least 3 correct terms or sight of 4.6 or correct $\text{Var}(X)$; NB $\frac{4.6}{5}$ loses M1 |
| $\text{Var}(X) = E(X^2) - 1.8^2 = 1.36$ or $\frac{34}{25}$ | M1 | Follow through their "4.6"; must see $-1.8^2$ or 1.36 |
| $\text{Var}(3-2X) = (-2)^2 \text{Var}(X)$ | M1 | Correct use of $\text{Var}(aX+b)$ formula; condone $-2^2$ if this later becomes $+4$ |
| $= 4 \times 1.36 = 5.44$ | A1 | Accept $\frac{136}{25}$ or exact equivalent |
---4. A discrete random variable $X$ has the probability distribution given in the table below, where $a$ and $b$ are constants.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & $a$ & $\frac { 1 } { 10 }$ & $\frac { 1 } { 5 }$ & $\frac { 3 } { 10 }$ & $b$ \\
\hline
\end{tabular}
\end{center}
Given $\mathrm { E } ( X ) = \frac { 9 } { 5 }$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item find two simultaneous equations for $a$ and $b$,
\item show that $a = \frac { 1 } { 20 }$ and find the value of $b$.
\end{enumerate}\item Specify the cumulative distribution function $\mathrm { F } ( x )$ for $x = - 1,0,1$, 2 and 3
\item Find $\mathrm { P } ( X < 2.5 )$.
\item Find $\operatorname { Var } ( 3 - 2 X )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-13_90_68_2613_1877}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2014 Q4 [11]}}