| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Moderate -0.3 This is a straightforward application of standard results for linear combinations of independent normal random variables. Parts (a)-(c) require direct use of E(aX+bY) and Var(aX+bY) formulas plus a normal probability calculation. Part (d) extends this to multiple variables but uses the same principles. All steps are routine with no problem-solving insight required, making it slightly easier than average for A-level. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks |
|---|---|
| \(E(R) = E(X) + 4E(Y) = 8 + (4 \times 14) = 64\) | M1 A1 |
| Total: 2 marks |
| Answer | Marks |
|---|---|
| \(\text{Var}(R) = \text{Var}(X) + 16\text{Var}(Y) = 2^2 + (16 \times 3^2)\) | M1 A1 |
| \(= 148\) | A1 |
| Total: 3 marks |
| Answer | Marks |
|---|---|
| \(P(R < 41) = P\left(Z < \frac{41 - 64}{\sqrt{148}}\right) = P(Z < -1.89)\) | M1 A1 \(\checkmark\) |
| \(= 0.0294\) | A1 |
| Total: 3 marks |
| Answer | Marks |
|---|---|
| \(\text{Var}(S) = 3\text{Var}(Y) + \left(\frac{1}{2}\right)^2 \text{Var}(X)\) | M1 M1 |
| \(= 27 + 1\) | A1 |
| \(= 28\) | A1 |
| Total: 4 marks |
## Part (a)
| $E(R) = E(X) + 4E(Y) = 8 + (4 \times 14) = 64$ | M1 A1 |
| **Total: 2 marks** |
## Part (b)
| $\text{Var}(R) = \text{Var}(X) + 16\text{Var}(Y) = 2^2 + (16 \times 3^2)$ | M1 A1 |
| $= 148$ | A1 |
| **Total: 3 marks** |
## Part (c)
| $P(R < 41) = P\left(Z < \frac{41 - 64}{\sqrt{148}}\right) = P(Z < -1.89)$ | M1 A1 $\checkmark$ |
| $= 0.0294$ | A1 |
| **Total: 3 marks** |
## Part (d)
| $\text{Var}(S) = 3\text{Var}(Y) + \left(\frac{1}{2}\right)^2 \text{Var}(X)$ | M1 M1 |
| $= 27 + 1$ | A1 |
| $= 28$ | A1 |
| **Total: 4 marks** |
---The random variable $R$ is defined as $R = X + 4Y$ where $X \sim \text{N}(8, 2^2)$, $Y \sim \text{N}(14, 3^2)$ and $X$ and $Y$ are independent.
Find
\begin{enumerate}[label=(\alph*)]
\item E$(R)$, [2]
\item Var$(R)$, [3]
\item P$(R < 41)$ [3]
\end{enumerate}
The random variables $Y_1$, $Y_2$ and $Y_3$ are independent and each has the same distribution as $Y$. The random variable $S$ is defined as
$$S = \sum_{i=1}^{3} Y_i - \frac{1}{2}X.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find Var$(S)$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q5 [12]}}