| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Discrete CDF to PMF |
| Difficulty | Moderate -0.8 This is a straightforward S1 question requiring basic understanding of CDFs and PMFs. Students need to recognize that F(3) = P(X=2) = a, F(6) = a+b, then use the probability sum equals 1 to find c. Part (b) is immediate recall. Requires only direct application of definitions with minimal problem-solving. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general |
| \(x\) | 2 | 4 | 7 | 10 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(b\) | 0.1 | \(c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F(3) = P(X=2)\) so \(a = 0.2\) | B1 | For \(a = 0.2\) |
| \(F(6) = P(X=2) + P(X=4)\) so \(a + b = 0.8\), so \(b = 0.6\) | B1 | For \(b = 0.6\) |
| Sum of probs \(= 1\) implies \(c = 0.1\) | B1ft | For \(c = 0.1\), or a value of \(c\) so that \(a+b+c=0.9\) provided \(a\), \(b\), \(c\) are probabilities. Labels may not be explicit but must be clear which is which. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F(7) = F(6) + 0.1\) or \(a+b+0.1\) or \(1-c = \mathbf{0.9}\) | B1 | For \(0.9\) only (no ft). If answer based on their \(a\), \(b\), \(c\), those values must be probabilities and have \(a+b=0.8\) or \(c=0.1\). Just stating \(0.9\) with no justification is B1. |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F(3) = P(X=2)$ so $a = 0.2$ | B1 | For $a = 0.2$ |
| $F(6) = P(X=2) + P(X=4)$ so $a + b = 0.8$, so $b = 0.6$ | B1 | For $b = 0.6$ |
| Sum of probs $= 1$ implies $c = 0.1$ | B1ft | For $c = 0.1$, or a value of $c$ so that $a+b+c=0.9$ provided $a$, $b$, $c$ are probabilities. Labels may not be explicit but must be clear which is which. |
**(3 marks)**
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F(7) = F(6) + 0.1$ or $a+b+0.1$ or $1-c = \mathbf{0.9}$ | B1 | For $0.9$ only (no ft). If answer based on their $a$, $b$, $c$, those values must be probabilities and have $a+b=0.8$ or $c=0.1$. Just stating $0.9$ with no justification is B1. |
**(1 mark) [Total: 4]**
---\begin{enumerate}
\item The discrete random variable $X$ has the following probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 2 & 4 & 7 & 10 \\
\hline
$\mathrm { P } ( X = x )$ & $a$ & $b$ & 0.1 & $c$ \\
\hline
\end{tabular}
\end{center}
where $a , b$ and $c$ are probabilities.\\
The cumulative distribution function of $X$ is $\mathrm { F } ( x )$ and $\mathrm { F } ( 3 ) = 0.2$ and $\mathrm { F } ( 6 ) = 0.8$\\
(a) Find the value of $a$, the value of $b$ and the value of $c$.\\
(b) Write down the value of $\mathrm { F } ( 7 )$.\\
\hfill \mbox{\textit{Edexcel S1 2018 Q1 [4]}}