| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Distribution of linear combination |
| Difficulty | Standard +0.8 This S3 question requires understanding of linear combinations of normal variables and conditional probability. Part (a) involves working backwards from a probability to find a parameter, requiring inverse normal calculations and variance rules. Part (b) combines multiple independent normals and applies conditional probability with standardization. While the techniques are standard for S3, the multi-step nature and need to correctly handle variance scaling (not standard deviation) makes this moderately challenging. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
6. The random variable $W$ is defined as
$$W = 3 X - 4 Y$$
where $X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)$ and $Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)$ and $X$ and $Y$ are independent.\\
Given that $\mathrm { P } ( W < 44 ) = 0.9$
\begin{enumerate}[label=(\alph*)]
\item find the value of $\sigma$, giving your answer to 2 decimal places.
The random variables $A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ each have the same distribution as $A$, where $A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)$
The random variable $B$ is defined as
$$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$
where $X , A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ are independent.
\item Find $\mathrm { P } ( B \leqslant 145 \mid B > 120 )$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2016 Q6 [15]}}