| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Comparison involving sums or multiples |
| Difficulty | Standard +0.3 This is a straightforward application of standard results for linear combinations of normal random variables. Parts (a) and (b) require direct use of E(aX + bY) and Var(aX + bY) formulas with no problem-solving. Part (c) involves finding P(B > A) = P(B - A > 0), requiring recognition that B - A is normal, then calculating its mean and variance before using tables—mechanical but slightly more involved than average. Overall slightly easier than a typical A-level question due to being purely procedural. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
The random variable $A$ is defined as
$$A = 4X - 3Y$$
where $X \sim \text{N}(30, 3^2)$, $Y \sim \text{N}(20, 2^2)$ and $X$ and $Y$ are independent.
Find
\begin{enumerate}[label=(\alph*)]
\item E($A$), [2]
\item Var($A$). [3]
\end{enumerate}
The random variables $Y_1$, $Y_2$, $Y_3$ and $Y_4$ are independent and each has the same distribution as $Y$. The random variable $B$ is defined as
$$B = \sum_{i=1}^{4} Y_i$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find P($B > A$). [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2009 Q8 [11]}}