| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2004 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.3 This is a straightforward application of standard results for linear combinations of independent normal random variables. Students need to find the mean and variance using E(aX+bY) = aE(X)+bE(Y) and Var(aX+bY) = a²Var(X)+b²Var(Y), then use normal tables. Part (b) adds minimal complexity by requiring recognition that the sum of three independent B variables has variance 3×3² = 27. This is slightly easier than average as it's purely procedural with no problem-solving or insight required. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
7. The random variable $D$ is defined as
$$D = A - 3 B + 4 C$$
where $A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)$ and $C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)$, and $A , B$ and $C$ are independent.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( \mathrm { D } < 44 )$.
The random variables $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ are independent and each has the same distribution as $B$. The random variable $X$ is defined as
$$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C .$$
\item Find $\mathrm { P } ( X > 0 )$.
\section*{END}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2004 Q7 [16]}}