| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Discrete CDF to PMF |
| Difficulty | Easy -1.2 This is a straightforward question testing basic understanding of probability distributions and CDFs. Part (a) requires simple arithmetic using the definitions F(y) = P(Y ≤ y) and ΣP = 1, with no problem-solving insight needed. Part (b) is routine inequality manipulation (3Y + 2 ≥ 8 gives Y ≥ 2) followed by adding probabilities. This is easier than average A-level content, being a standard S1 bookwork exercise with clear mechanical steps. |
| Spec | 5.02a Discrete probability distributions: general |
| \(y\) | 1 | 2 | 3 | 4 |
| \(\text{P}(Y = y)\) | \(a\) | \(b\) | 0.3 | \(c\) |
| \(y\) | 1 | 2 | 3 | 4 |
| F(\(y\)) | 0.1 | 0.5 | \(d\) | 1.0 |
| Answer | Marks | Guidance |
|---|---|---|
| [F(3) = F(2) + P(Y =3) = (0.5 + 0.3)] | B1, B1 | |
| \(b = \text{F(2)} - a = 0.5 - 0.1\) or \(a + b = 0.5\) | M1 | |
| \(c = 1 - \text{F(3)}\) or \(1 - (a + b + 0.3)\) or \(a + b + c = 0.7\) | A1, A1 | |
| \(a = 0.1\) | ||
| \(d = 0.8\) | ||
| \(b = 0.4\) | ||
| \(c = 0.2\) | Correct answers with no (or irrelevant) working score full marks |
| Answer | Marks | Guidance |
|---|---|---|
| P(\(3Y + 2 \geq 8\)) = P(\(Y \geq 2\)) or \(1 - \text{P}(Y \leq 1)\) or \(1 - a\) = 0.9 | M1, A1ft | For rearranging to P(\(Y \geq 2\)) or 1 − P(\(Y \leq 1\)) or selecting cases \(Y = 2, 3\) and 4. For 0.3 + their b + their c or 1 - their a, provided final answer < 1 and their values are probabilities. |
## (a)
[F(3) = F(2) + P(Y =3) = (0.5 + 0.3)] | B1, B1 |
$b = \text{F(2)} - a = 0.5 - 0.1$ or $a + b = 0.5$ | M1 |
$c = 1 - \text{F(3)}$ or $1 - (a + b + 0.3)$ or $a + b + c = 0.7$ | A1, A1 |
$a = 0.1$ | |
$d = 0.8$ | |
$b = 0.4$ | |
$c = 0.2$ | | Correct answers with no (or irrelevant) working score full marks
## (b)
P($3Y + 2 \geq 8$) = P($Y \geq 2$) or $1 - \text{P}(Y \leq 1)$ or $1 - a$ = 0.9 | M1, A1ft | For rearranging to P($Y \geq 2$) or 1 − P($Y \leq 1$) or selecting cases $Y = 2, 3$ and 4. For 0.3 + their b + their c or 1 - their a, provided final answer < 1 and their values are probabilities.
---The discrete random variable $Y$ has probability distribution
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$y$ & 1 & 2 & 3 & 4 \\
\hline
$\text{P}(Y = y)$ & $a$ & $b$ & 0.3 & $c$ \\
\hline
\end{tabular}
\end{center}
where $a$, $b$ and $c$ are constants.
The cumulative distribution function F($y$) of $Y$ is given in the following table
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$y$ & 1 & 2 & 3 & 4 \\
\hline
F($y$) & 0.1 & 0.5 & $d$ & 1.0 \\
\hline
\end{tabular}
\end{center}
where $d$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$, the value of $b$, the value of $c$ and the value of $d$. [5]
\item Find $\text{P}(3Y + 2 \geq 8)$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2011 Q3 [7]}}