| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Discrete CDF to PMF |
| Difficulty | Easy -1.2 This is a straightforward S1 question testing basic understanding of the relationship between PMF and CDF. Part (a) requires simple arithmetic using F(x) = P(X≤x) and probabilities summing to 1. Part (b) is direct identification of values. No problem-solving or novel insight needed—purely routine application of definitions. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | - 1 | 0 | 1 | 2 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(b\) | \(b\) | \(c\) |
| \(x\) | - 1 | 0 | 1 | 2 |
| \(\mathrm {~F} ( x )\) | \(\frac { 1 } { 3 }\) | \(d\) | \(\frac { 5 } { 6 }\) | \(e\) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(a = \frac{1}{3}\) and \(e = 1\) | B1 |
| \(c = [1 - \frac{1}{6}] = \frac{1}{6}\) | B1 | |
| \("·\frac{1}{3}"+ 2b = \frac{5}{6}\) or \("·\frac{1}{3}" + 2b + "·\frac{1}{6}" = 1\) | M1 | |
| \(⟹ b = \frac{1}{4}\) | A1 | |
| \(d = a + b = "·\frac{1}{3}" + "·\frac{1}{4}"\) or \(d = \frac{5}{6} - "·\frac{1}{4}"\) (o.e.) so \(d = \frac{7}{12}\) | B1ft | |
| (b) | \([P(X^2 = 1) = a + b] = \frac{7}{12}\) | B1ft |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Probabilities not in \([0, 1]\) score 0 for corresponding A or B marks. Allow exact decimals or equivalent fractions. In part (a) you may see answers in the tables. If answers in the table and answers on the page disagree take the answers on the page. If jumbled working is followed by a list of answers on the page mark the list. | M1 for an equation for \(b\). Follow through their value of \(a\) and possibly \(c\) if both in \([0,1]\). Must be seen as an equation with \(b\) the only unknown. NB \(b = d - a\) is not a suitable equation and use of this is M0. 1st A1 for \(b = \frac{1}{4}\) or 0.25 (Correct answer only is 2/2). 3rd B1ft for \(d = \frac{7}{12}\) or their \(a +\) their \(b\) but their \(d\) must satisfy \(\frac{1}{3} < d < \frac{5}{6}\) |
| (b) | B1ft for \(\frac{7}{12}\) or their \(a +\) their \(b\) or their \(d\) |
(a) | $a = \frac{1}{3}$ and $e = 1$ | B1 | |
| $c = [1 - \frac{1}{6}] = \frac{1}{6}$ | B1 | |
| $"·\frac{1}{3}"+ 2b = \frac{5}{6}$ or $"·\frac{1}{3}" + 2b + "·\frac{1}{6}" = 1$ | M1 | |
| $⟹ b = \frac{1}{4}$ | A1 | |
| $d = a + b = "·\frac{1}{3}" + "·\frac{1}{4}"$ or $d = \frac{5}{6} - "·\frac{1}{4}"$ (o.e.) so $d = \frac{7}{12}$ | B1ft | |
(b) | $[P(X^2 = 1) = a + b] = \frac{7}{12}$ | B1ft | For $\frac{7}{12}$ or their $a +$ their $b$ or their $d$ |
**Notes:**
(a) | Probabilities not in $[0, 1]$ score 0 for corresponding A or B marks. Allow exact decimals or equivalent fractions. In part (a) you may see answers in the tables. If answers in the table and answers on the page disagree take the answers on the page. If jumbled working is followed by a list of answers on the page mark the list. | M1 for an equation for $b$. Follow through their value of $a$ and possibly $c$ if both in $[0,1]$. Must be seen as an equation with $b$ the only unknown. NB $b = d - a$ is not a suitable equation and use of this is M0. 1st A1 for $b = \frac{1}{4}$ or 0.25 (Correct answer only is 2/2). 3rd B1ft for $d = \frac{7}{12}$ or their $a +$ their $b$ but their $d$ must satisfy $\frac{1}{3} < d < \frac{5}{6}$ |
(b) | B1ft for $\frac{7}{12}$ or their $a +$ their $b$ or their $d$ |4. The discrete random variable $X$ has probability distribution
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 2 \\
\hline
$\mathrm { P } ( X = x )$ & $a$ & $b$ & $b$ & $c$ \\
\hline
\end{tabular}
\end{center}
The cumulative distribution function of $X$ is given by
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 2 \\
\hline
$\mathrm {~F} ( x )$ & $\frac { 1 } { 3 }$ & $d$ & $\frac { 5 } { 6 }$ & $e$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a , b , c , d$ and $e$.
\item Write down the value of $\mathrm { P } \left( X ^ { 2 } = 1 \right)$.\\
\section*{}
\section*{\begin{center}
\includegraphics[max width=\textwidth, alt={}]{image-not-found}
\end{center}}
$T$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2017 Q4 [6]}}