| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Easy -1.2 This is a straightforward discrete probability distribution question requiring only routine application of standard formulas. Students must find k using ΣP(X=x)=1, construct a table, then calculate E(X) and Var(X) using standard definitions—all mechanical calculations with no problem-solving or conceptual insight required. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = \frac{1}{18}\) \((4k + k + 4k + 9k = 18k = 1)\) | B1 | SOI |
| Table with \(x\) values \(-2, 1, 2, 3\) and \(P(X=x)\) values \(\frac{4}{18}, \frac{1}{18}, \frac{4}{18}, \frac{9}{18}\) | M1 | Table with correct \(x\) values and at least one probability accurate using their \(k\). Values need not be in order, lines may not be drawn, may be vertical, \(x\) and \(P(X=x)\) may be omitted. Condone any additional \(X\) values if probability stated as 0. |
| Remaining probabilities correct | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[\text{E}(X) = \frac{4 \times -2 + 1 \times 1 + 4 \times 2 + 9 \times 3}{18} = \right] \frac{-8+1+8+27}{18}\) | M1 | \(-8k + k + 8k + 27k\). May be implied by use in Variance. Accept unsimplified expression. FT their table if probabilities sum to 1 or 0.999. SC B1 \(28k\) |
| \(\left[\text{Var}(X) = \frac{4 \times (-2)^2 + 1 \times 1^2 + 4 \times 2^2 + 9 \times 3^2}{18} - (\text{their } \text{E}(X))^2 = \right]\) \(= \frac{16+1+16+81}{18} - \left(\text{their } \frac{28}{18}\right)^2\) | M1 | \(16k + k + 16k + 81k - (\text{their mean})^2\). FT their table even if probabilities not summing to 1. Note: If table is correct, \(\frac{114}{18} - (\text{their } \text{E}(X))^2\) M1. SC B1 \(114k - (\text{their mean})^2\) |
| \(\text{E}(X) = \frac{14}{9}, 1\frac{5}{9}, 1.56\), \(\text{Var}(X) = \frac{317}{81}, 3\frac{74}{81}, 3.91\) | A1 | Answers for \(\text{E}(X)\) and \(\text{Var}(X)\) must be identified. \(3.91 \leqslant \text{Var}(X) \leqslant 3.914\) |
## Question 3:
**Part 3(a):**
$k = \frac{1}{18}$ $(4k + k + 4k + 9k = 18k = 1)$ | B1 | SOI
Table with $x$ values $-2, 1, 2, 3$ and $P(X=x)$ values $\frac{4}{18}, \frac{1}{18}, \frac{4}{18}, \frac{9}{18}$ | M1 | Table with correct $x$ values and at least one probability accurate using their $k$. Values need not be in order, lines may not be drawn, may be vertical, $x$ and $P(X=x)$ may be omitted. Condone any additional $X$ values if probability stated as 0.
Remaining probabilities correct | A1 |
**Part 3(b):**
$\left[\text{E}(X) = \frac{4 \times -2 + 1 \times 1 + 4 \times 2 + 9 \times 3}{18} = \right] \frac{-8+1+8+27}{18}$ | M1 | $-8k + k + 8k + 27k$. May be implied by use in Variance. Accept unsimplified expression. FT their table if probabilities sum to 1 or 0.999. **SC B1** $28k$
$\left[\text{Var}(X) = \frac{4 \times (-2)^2 + 1 \times 1^2 + 4 \times 2^2 + 9 \times 3^2}{18} - (\text{their } \text{E}(X))^2 = \right]$ $= \frac{16+1+16+81}{18} - \left(\text{their } \frac{28}{18}\right)^2$ | M1 | $16k + k + 16k + 81k - (\text{their mean})^2$. FT their table even if probabilities not summing to 1. Note: If table is correct, $\frac{114}{18} - (\text{their } \text{E}(X))^2$ M1. **SC B1** $114k - (\text{their mean})^2$
$\text{E}(X) = \frac{14}{9}, 1\frac{5}{9}, 1.56$, $\text{Var}(X) = \frac{317}{81}, 3\frac{74}{81}, 3.91$ | A1 | Answers for $\text{E}(X)$ and $\text{Var}(X)$ must be identified. $3.91 \leqslant \text{Var}(X) \leqslant 3.914$
---3 The random variable $X$ takes the values $- 2,1,2,3$. It is given that $\mathrm { P } ( X = x ) = k x ^ { 2 }$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions.
\item Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q3 [6]}}