| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution question requiring only routine application of ΣP(X=x)=1 to find k, then standard formulas for expectation and variance. The algebraic manipulation is minimal (substituting three values into x²-1), and all techniques are basic S1 content with no problem-solving insight needed. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([3k + 3k + 8k = 1, \text{ so}]\ k = \dfrac{1}{14}\) | B1 | |
| Table with correct values of \(x\): \(-2, 2, 3\) and probabilities \(\dfrac{3}{14}\ (0.214),\ \dfrac{3}{14}\ (0.214),\ \dfrac{8}{14}\ (0.571)\) | B1 FT | Table with correct values of \(x\), and at least one correct probability linked with outcome. FT *their* \(k\). Condone any additional \(X\) values if probability stated as \(0\). |
| B1 FT | The outcomes in the table must be \(-2\), \(2\) and \(3\). 2 further correct probabilities in table or 3 correct probabilities not in table linked to outcomes, or 3 correct FT probabilities in table using *their* \(k\), or 3 incorrect probabilities summing to \(1\) in table if \(k\) not stated. | |
| If \(k\) not calculated, SC B1 for table with \(x\): \(-2, 2, 3\) and \(\text{P}(x)\): \(3k,\ 3k,\ 8k\) | ||
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = -2 \times \frac{3}{14} + 2 \times \frac{3}{14} + 3 \times \frac{8}{14}\) | M1 | Accept unsimplified expression. May be calculated in variance. FT their table with 3 probabilities summing to \(0.999 \leq total \leq 1\) \((0 < p < 1)\) or in terms of \(k\) |
| \(-\frac{6}{14} + \frac{6}{14} + \frac{24}{14}\) | ||
| \(Var(X) = (-2)^2 \times \frac{3}{14} + 2^2 \times \frac{3}{14} + 3^2 \times \frac{8}{14} - (their\ E(X))^2\) | M1 | Appropriate variance formula using their \((E(X))^2\) value. FT their table with 3 or more probabilities \((0 < p < 1)\) which need not sum to 1, or in terms of \(k\) with an expression no more evaluated than shown |
| \(4 \times \frac{3}{14} + 4 \times \frac{3}{14} + 9 \times \frac{8}{14} - \left(their\ \frac{12}{7}\right)^2\) | ||
| \(\left[\frac{12+12+72}{14} - \left(their\ \frac{12}{7}\right)^2\right]\) | ||
| \(E(X) = \frac{12}{7},\ 1.71,\ 1\frac{5}{7}\) | A1 | Answers for \(E(X)\) and \(Var(X)\) must be identified. \(E(X)\) may be identified by correct use in Variance (condone E, V, \(\mu\), \(\sigma^2\), etc.). If A0 earned, SC B1 for identified correct final answers |
| \(Var(X) = \frac{192}{49},\ 3.92,\ 3\frac{45}{49}\) | ||
| Total | 3 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[3k + 3k + 8k = 1, \text{ so}]\ k = \dfrac{1}{14}$ | **B1** | |
| Table with correct values of $x$: $-2, 2, 3$ and probabilities $\dfrac{3}{14}\ (0.214),\ \dfrac{3}{14}\ (0.214),\ \dfrac{8}{14}\ (0.571)$ | **B1 FT** | Table with correct values of $x$, and at least one correct probability linked with outcome. FT *their* $k$. Condone any additional $X$ values if probability stated as $0$. |
| | **B1 FT** | The outcomes in the table must be $-2$, $2$ and $3$. 2 further correct probabilities in table **or** 3 correct probabilities not in table linked to outcomes, **or** 3 correct FT probabilities in table using *their* $k$, **or** 3 incorrect probabilities summing to $1$ in table if $k$ not stated. |
| | | If $k$ not calculated, **SC B1** for table with $x$: $-2, 2, 3$ and $\text{P}(x)$: $3k,\ 3k,\ 8k$ |
| **Total: 3** | | |
## Question 1:
**Part (b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = -2 \times \frac{3}{14} + 2 \times \frac{3}{14} + 3 \times \frac{8}{14}$ | M1 | Accept unsimplified expression. May be calculated in variance. FT their table with 3 probabilities summing to $0.999 \leq total \leq 1$ $(0 < p < 1)$ or in terms of $k$ |
| $-\frac{6}{14} + \frac{6}{14} + \frac{24}{14}$ | | |
| $Var(X) = (-2)^2 \times \frac{3}{14} + 2^2 \times \frac{3}{14} + 3^2 \times \frac{8}{14} - (their\ E(X))^2$ | M1 | Appropriate variance formula using their $(E(X))^2$ value. FT their table with 3 or more probabilities $(0 < p < 1)$ which need not sum to 1, or in terms of $k$ with an expression no more evaluated than shown |
| $4 \times \frac{3}{14} + 4 \times \frac{3}{14} + 9 \times \frac{8}{14} - \left(their\ \frac{12}{7}\right)^2$ | | |
| $\left[\frac{12+12+72}{14} - \left(their\ \frac{12}{7}\right)^2\right]$ | | |
| $E(X) = \frac{12}{7},\ 1.71,\ 1\frac{5}{7}$ | A1 | Answers for $E(X)$ and $Var(X)$ must be identified. $E(X)$ may be identified by correct use in Variance (condone E, V, $\mu$, $\sigma^2$, etc.). If A0 earned, **SC B1** for identified correct final answers |
| $Var(X) = \frac{192}{49},\ 3.92,\ 3\frac{45}{49}$ | | |
| **Total** | **3** | |
---1 The random variable $X$ takes the values $- 2,2$ and 3. It is given that
$$\mathrm { P } ( X = x ) = k \left( x ^ { 2 } - 1 \right)$$
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 2023 Q1 [6]}}