| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | November |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Easy -1.3 This is a straightforward S1 question requiring only basic probability axioms (probabilities sum to 1) and standard variance formula application. Part (i) is simple algebra (p + p + 2p + 2p + 0.1 = 1), and part (ii) is direct substitution into Var(X) = E(X²) - [E(X)]² with E(X) given. No problem-solving insight needed, just routine application of definitions. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | - 1 | 0 | 1 | 2 | 4 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(p\) | \(2 p\) | \(2 p\) | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(6p + 0.1 = 1\), \(p = 0.15\) | B1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Var}(X) = 1 \times p + 1 \times 2p + 4 \times 2p + 16 \times 0.1 - 1.15^2\) | M1 | Correct unsimplified formula, *their p* substituted (allow 1 error) |
| \(0.15 + 0 + 0.3 + 1.2 + 1.6 - 1.15^2 = 1.9275 = 1.93\) (3sf) | A1 | Correct answer |
**Question 2(i):**
$6p + 0.1 = 1$, $p = 0.15$ | B1 | Correct answer
**Total: 1 mark**
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**Question 2(ii):**
$\text{Var}(X) = 1 \times p + 1 \times 2p + 4 \times 2p + 16 \times 0.1 - 1.15^2$ | M1 | Correct unsimplified formula, *their p* substituted (allow 1 error)
$0.15 + 0 + 0.3 + 1.2 + 1.6 - 1.15^2 = 1.9275 = 1.93$ (3sf) | A1 | Correct answer
**Total: 2 marks**2 A random variable $X$ has the probability distribution shown in the following table, where $p$ is a constant.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 2 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & $p$ & $p$ & $2 p$ & $2 p$ & 0.1 \\
\hline
\end{tabular}
\end{center}
(i) Find the value of $p$.\\
(ii) Given that $\mathrm { E } ( X ) = 1.15$, find $\operatorname { Var } ( X )$.\\
\hfill \mbox{\textit{CAIE S1 2018 Q2 [3]}}