| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Distribution of linear combination |
| Difficulty | Standard +0.3 This is a standard S3 question on linear combinations of normal variables requiring routine application of formulas for mean and variance of linear combinations. Part (a) is straightforward calculation, while part (b) requires working backwards from a probability to find σ, involving one additional step of algebraic manipulation. The techniques are well-practiced at this level with no novel insight required. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(E(X) = 4 \times 15 - 3 \times 10 = 30\) | M1 | For a correct method to find \(E(X)\). May be implied by a correct standardisation expression. |
| \(\text{Var}(X) = 4^2 \times 5^2 + 3^2 \times 4^2 = 544\) | M1 | For a correct method to find \(\text{Var}(X)\). Allow \(\sqrt{544}\) or \(23.3^2\) or better. May be implied by a correct standardisation expression. |
| So \(X \sim N(30, 544)\) | ||
| \(P(X < 40) = P\left(Z < \frac{40 - 30}{\sqrt{544}}\right) = P(Z < 0.428...)\) | M1 | For standardising \((\pm)\) using their mean and their variance |
| \(= 0.6664\) (Calculator gives \(0.6659...\)) awrt \(0.666\) | A1 | awrt 0.666 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(E(A+B+D) = 15 + 10 + 3 \times 20 = 85\) | M1 | For a correct method to find \(E(A+B+D)\) |
| \(\text{Var}(A+B+D) = 5^2 + 4^2 + 3 \times \sigma^2 = 41 + 3\sigma^2\) | M1 | For a correct method to find \(\text{Var}(A+B+D)\) |
| So \(A+B+D \sim N(85,\ 41+3\sigma^2)\) | ||
| \(P(A+B+D < 76) = P\left(Z < \frac{76-85}{\sqrt{41+3\sigma^2}}\right) = 0.242\) | ||
| \(\frac{-9}{\sqrt{41+3\sigma^2}} = -0.7\) or \(\frac{9}{\sqrt{41+3\sigma^2}} = 0.7\) (Calculator gives \(-0.69988...\)) | M1 A1 | For standardising \((\pm)\) using their mean and their standard deviation which is in terms of \(\sigma^2\) and setting equal to \(-0.7\) or better. Allow \(+0.7\). A1 for the correct equation. |
| \(3\sigma^2 = \left(\frac{-9}{-0.7}\right)^2 - 41\) | dM1 | Dependent on previous M mark. For squaring and rearranging leading to an equation in \(\sigma^2\) |
| \(\sigma = 6.437...\) awrt \(6.44\) | A1 | awrt 6.44 (Do not award if previous A mark was not awarded) |
## Question 7:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $E(X) = 4 \times 15 - 3 \times 10 = 30$ | M1 | For a correct method to find $E(X)$. May be implied by a correct standardisation expression. |
| $\text{Var}(X) = 4^2 \times 5^2 + 3^2 \times 4^2 = 544$ | M1 | For a correct method to find $\text{Var}(X)$. Allow $\sqrt{544}$ or $23.3^2$ or better. May be implied by a correct standardisation expression. |
| So $X \sim N(30, 544)$ | | |
| $P(X < 40) = P\left(Z < \frac{40 - 30}{\sqrt{544}}\right) = P(Z < 0.428...)$ | M1 | For standardising $(\pm)$ using their mean and their variance |
| $= 0.6664$ (Calculator gives $0.6659...$) awrt $0.666$ | A1 | awrt 0.666 |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $E(A+B+D) = 15 + 10 + 3 \times 20 = 85$ | M1 | For a correct method to find $E(A+B+D)$ |
| $\text{Var}(A+B+D) = 5^2 + 4^2 + 3 \times \sigma^2 = 41 + 3\sigma^2$ | M1 | For a correct method to find $\text{Var}(A+B+D)$ |
| So $A+B+D \sim N(85,\ 41+3\sigma^2)$ | | |
| $P(A+B+D < 76) = P\left(Z < \frac{76-85}{\sqrt{41+3\sigma^2}}\right) = 0.242$ | | |
| $\frac{-9}{\sqrt{41+3\sigma^2}} = -0.7$ or $\frac{9}{\sqrt{41+3\sigma^2}} = 0.7$ (Calculator gives $-0.69988...$) | M1 A1 | For standardising $(\pm)$ using their mean and their standard deviation which is in terms of $\sigma^2$ and setting equal to $-0.7$ or better. Allow $+0.7$. A1 for the correct equation. |
| $3\sigma^2 = \left(\frac{-9}{-0.7}\right)^2 - 41$ | dM1 | Dependent on previous M mark. For squaring and rearranging leading to an equation in $\sigma^2$ |
| $\sigma = 6.437...$ awrt $6.44$ | A1 | awrt 6.44 (Do not award if previous A mark was not awarded) |\begin{enumerate}
\item The random variable $X$ is defined as
\end{enumerate}
$$X = 4 A - 3 B$$
where $A$ and $B$ are independent and
$$A \sim \mathrm {~N} \left( 15,5 ^ { 2 } \right) \quad B \sim \mathrm {~N} \left( 10,4 ^ { 2 } \right)$$
(a) Find $\mathrm { P } ( X < 40 )$
The random variable $C$ is such that $C \sim \mathrm {~N} \left( 20 , \sigma ^ { 2 } \right)$\\
The random variables $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ are independent and each has the same distribution as $C$
The random variable $D$ is defined as
$$D = \sum _ { i = 1 } ^ { 3 } C _ { i }$$
Given that $\mathrm { P } ( A + B + D < 76 ) = 0.2420$ and that $A , B$ and $D$ are independent,\\
(b) showing your working clearly, find the standard deviation of $C$
\hfill \mbox{\textit{Edexcel S3 2023 Q7 [10]}}