| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Linear combinations of independent variables |
| Difficulty | Moderate -0.8 This question tests straightforward application of standard results for linear combinations of independent random variables (variance of sum, variance of linear transformation). All parts require direct formula application with no problem-solving or conceptual insight—purely mechanical calculation of Var(S) = 3×3.1, Var(T) = 9×3.1, and combining these for S-T. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(9.3\) | B1 [1] |
| (ii) | \(27.9\) | B1 [1] |
| (iii) | \(E(S) = 17.4\), \(E(T) = 19.4\); \(E(S-T) = -2.0\); \(\text{Var}(S-T) = 37.2\) | M1, A1, B1ft [3] |
## Question 1:
**(i)** | $9.3$ | B1 [1] |
**(ii)** | $27.9$ | B1 [1] |
**(iii)** | $E(S) = 17.4$, $E(T) = 19.4$; $E(S-T) = -2.0$; $\text{Var}(S-T) = 37.2$ | M1, A1, B1ft [3] | For subtracting $E[S]-E[T]$, can be non-numerical; ft (i) & (ii) adding (i) and (ii) ft non-negative answers only
---1 The mean and variance of the random variable $X$ are 5.8 and 3.1 respectively. The random variable $S$ is the sum of three independent values of $X$. The independent random variable $T$ is defined by $T = 3 X + 2$.\\
(i) Find the variance of $S$.\\
(ii) Find the variance of $T$.\\
(iii) Find the mean and variance of $S - T$.
\hfill \mbox{\textit{CAIE S2 2013 Q1 [5]}}