| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Probability distributions with parameters |
| Difficulty | Standard +0.8 This Further Statistics question requires setting up and solving a system involving the probability sum constraint, calculating E(X) and Var(X) with algebraic parameters, then using the condition σ=μ to find r. It demands careful algebraic manipulation with fractions and solving a non-trivial equation, going beyond routine probability calculations. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | \(r\) | \(k\) | \(\frac { k } { 2 }\) | \(\frac { k } { 3 }\) | \(\frac { k } { 4 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(X) = 0 \times r + k + 2 \times \frac{k}{2} + 3 \times \frac{k}{3} + 4 \times \frac{k}{4}\) or \(4k\) | B1 | For a correct expression for \(E(X)\) |
| \(E(X^2) = k + 2^2 \times \frac{k}{2} + 3^2 \times \frac{k}{3} + 4^2 \times \frac{k}{4}\) or \(k + 2k + 3k + 4k\) or \(10k\) | M1 | For attempting \(E(X^2)\) — at least 3 correct non-zero terms |
| \(\sqrt{\text{Var}(X)} = E(X) \Rightarrow E(X^2) = 2[E(X)]^2\) or \(10k - (4k)^2 = (4k)^2\) | M1 | Use of \(\text{Var}(X) = E(X^2) - [E(X)]^2\) to form an expression in \(k\) with \(\sqrt{\text{Var}(X)} = E(X)\); ft their \(10k\) and \(4k\); must use consistent expression for \(E(X)\) on both sides |
| \(10k = 32k^2 \Rightarrow k = \dfrac{5}{16}\) | A1 | For a correct equation for \(k\) (e.g. 2TQ) or \(k = \dfrac{5}{16}\) or \(0.3125\); may ignore or reject \(k = 0\) |
| \(\left[\sum \text{probs} = 1 \Rightarrow\right] r + \dfrac{1}{12}(12k + 6k + 4k + 3k) = 1\) | M1 | For attempt at an equation in \(r\) and \(k\) using sum of probs; at least 4 terms correct in terms of \(k\) or numerically using their value of \(k\) |
| \(r = 1 - \dfrac{25}{12} \times \dfrac{5}{16} = \dfrac{67}{192}\) | A1 | For \(\dfrac{67}{192}\) or exact equivalents e.g. \(0.3489583\); correct exact answer implies full marks |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(X) = 0 \times r + k + 2 \times \frac{k}{2} + 3 \times \frac{k}{3} + 4 \times \frac{k}{4}$ or $4k$ | B1 | For a correct expression for $E(X)$ |
| $E(X^2) = k + 2^2 \times \frac{k}{2} + 3^2 \times \frac{k}{3} + 4^2 \times \frac{k}{4}$ or $k + 2k + 3k + 4k$ or $10k$ | M1 | For attempting $E(X^2)$ — at least 3 correct non-zero terms |
| $\sqrt{\text{Var}(X)} = E(X) \Rightarrow E(X^2) = 2[E(X)]^2$ or $10k - (4k)^2 = (4k)^2$ | M1 | Use of $\text{Var}(X) = E(X^2) - [E(X)]^2$ to form an expression in $k$ with $\sqrt{\text{Var}(X)} = E(X)$; ft their $10k$ and $4k$; must use consistent expression for $E(X)$ on both sides |
| $10k = 32k^2 \Rightarrow k = \dfrac{5}{16}$ | A1 | For a correct equation for $k$ (e.g. 2TQ) or $k = \dfrac{5}{16}$ or $0.3125$; may ignore or reject $k = 0$ |
| $\left[\sum \text{probs} = 1 \Rightarrow\right] r + \dfrac{1}{12}(12k + 6k + 4k + 3k) = 1$ | M1 | For attempt at an equation in $r$ and $k$ using sum of probs; at least 4 terms correct in terms of $k$ or numerically using their value of $k$ |
| $r = 1 - \dfrac{25}{12} \times \dfrac{5}{16} = \dfrac{67}{192}$ | A1 | For $\dfrac{67}{192}$ or exact equivalents e.g. $0.3489583$; correct exact answer implies full marks |
**Total: (6 marks)**\begin{enumerate}
\item The discrete random variable $X$ has the following distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & $r$ & $k$ & $\frac { k } { 2 }$ & $\frac { k } { 3 }$ & $\frac { k } { 4 }$ \\
\hline
\end{tabular}
\end{center}
where $r$ and $k$ are positive constants.\\
The standard deviation of $X$ equals the mean of $X$\\
Find the exact value of $r$
\hfill \mbox{\textit{Edexcel FS1 AS 2023 Q1 [6]}}