| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| 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 variables. Students need to find E(A) and Var(A) using the formulas for linear combinations, then standardize and use tables. It's slightly easier than average because it's a direct one-step application with no problem-solving required beyond recalling the formulas. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(A) = E(B) + 4E(C) - 3E(D)\) | M1 | M1 for \(E(4C) = 4E(C)\) and \(-E(3D) = -3E(D)\) |
| \(= 22\) | A1 | A1 for 22 cao |
| \(\text{Var}(A) = \text{Var}(B) + 16\text{Var}(C) + 9\text{Var}(D)\) | M1 | M1 for use of \(\text{Var}(aX) = a^2\text{Var}(X)\) and adding their \(9\text{Var}(D)\) |
| \(= 168.25\) | A1 | A1 for 168.25 cao |
| \(P(A < 45) = P\left(Z < \frac{45-22}{\sqrt{168.25}}\right) = P(Z < 1.773)\) | M1 | M1 for standardising using their mean and their sd |
| \(= 0.9616\) | A1 | A1 awrt 0.962; calculator gives 0.961899… |
# Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(A) = E(B) + 4E(C) - 3E(D)$ | M1 | M1 for $E(4C) = 4E(C)$ **and** $-E(3D) = -3E(D)$ |
| $= 22$ | A1 | A1 for 22 cao |
| $\text{Var}(A) = \text{Var}(B) + 16\text{Var}(C) + 9\text{Var}(D)$ | M1 | M1 for use of $\text{Var}(aX) = a^2\text{Var}(X)$ and adding their $9\text{Var}(D)$ |
| $= 168.25$ | A1 | A1 for 168.25 cao |
| $P(A < 45) = P\left(Z < \frac{45-22}{\sqrt{168.25}}\right) = P(Z < 1.773)$ | M1 | M1 for standardising using their mean and their sd |
| $= 0.9616$ | A1 | A1 awrt 0.962; calculator gives 0.961899… |
---\begin{enumerate}
\item The random variable $A$ is defined as
\end{enumerate}
$$A = B + 4 C - 3 D$$
where $B$, $C$ and $D$ are independent random variables with
$$B \sim \mathrm {~N} \left( 6,2 ^ { 2 } \right) \quad C \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right) \quad D \sim \mathrm {~N} \left( 4,1.5 ^ { 2 } \right)$$
Find $\mathrm { P } ( A < 45 )$\\
\hfill \mbox{\textit{Edexcel S3 2014 Q4 [6]}}