| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2024 |
| 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 question requires setting up and solving simultaneous equations from probability axioms and expectation, then computing E(√(X+1)) which involves manipulating surds. While systematic, it demands careful algebraic manipulation across multiple steps, parameter solving, and expressing the final answer in exact surd form—significantly more demanding than routine expectation calculations but not requiring deep conceptual insight. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | - 1 | 0 | 1 | 3 | 7 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(r\) | \(p\) | 0.3 | \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([E(X) =]\ -1 \times p + 0 + 1 \times p + 3 \times 0.3 + 7r\ \text{or}\ 0.9 + 7r\ [= 1.95]\) | M1 | For an attempt at an expression for \(E(X)\), at least 3 correct non-zero products |
| \([7r = 1.05\ \text{so}]\ \mathbf{r = 0.15}\) | A1 | For \(r = 0.15\) oe |
| \([\text{Sum of probs} = 1]\ 2p + 0.3 + 2r = 1\) | M1 | For using sum of probabilities \(= 1\) to form an equation for \(p\) (ft their value or letter \(r\)) |
| \(\mathbf{p = 0.2}\) | A1ft | For \(p = 0.2\) oe or ft their \(r\) for \(p = \dfrac{0.7 - 2 \times \text{"0.15"}}{2}\) (provided \(p\) is a probability) |
| \([\text{Let}\ Y = \sqrt{X+1}]\) table of \(y\) values: \(0, 1, \sqrt{2}, 2, \sqrt{8}\ \text{or}\ 2\sqrt{2}\) with probabilities "0.2", "0.15", "0.2", 0.3, "0.15" | — | — |
| \([E(Y) =]\ 0 + \text{"0.15"} + \text{"0.2"}\sqrt{2} + 2 \times 0.3 + \text{"0.15"} \times \sqrt{8}\) | M1 | For attempt at \(E(Y)\) with at least 2 correct \(y\) values and products ft their \(p\) and \(r\) |
| \(= \mathbf{0.75 + 0.5\sqrt{2}}\) | A1cao (6) | cao; allow fraction or decimal, also allow \(\dfrac{3 + 2\sqrt{2}}{4}\) |
# Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| $[E(X) =]\ -1 \times p + 0 + 1 \times p + 3 \times 0.3 + 7r\ \text{or}\ 0.9 + 7r\ [= 1.95]$ | M1 | For an attempt at an expression for $E(X)$, at least 3 correct non-zero products |
| $[7r = 1.05\ \text{so}]\ \mathbf{r = 0.15}$ | A1 | For $r = 0.15$ oe |
| $[\text{Sum of probs} = 1]\ 2p + 0.3 + 2r = 1$ | M1 | For using sum of probabilities $= 1$ to form an equation for $p$ (ft their value or letter $r$) |
| $\mathbf{p = 0.2}$ | A1ft | For $p = 0.2$ oe or ft their $r$ for $p = \dfrac{0.7 - 2 \times \text{"0.15"}}{2}$ (provided $p$ is a probability) |
| $[\text{Let}\ Y = \sqrt{X+1}]$ table of $y$ values: $0, 1, \sqrt{2}, 2, \sqrt{8}\ \text{or}\ 2\sqrt{2}$ with probabilities "0.2", "0.15", "0.2", 0.3, "0.15" | — | — |
| $[E(Y) =]\ 0 + \text{"0.15"} + \text{"0.2"}\sqrt{2} + 2 \times 0.3 + \text{"0.15"} \times \sqrt{8}$ | M1 | For attempt at $E(Y)$ with at least 2 correct $y$ values and products ft their $p$ and $r$ |
| $= \mathbf{0.75 + 0.5\sqrt{2}}$ | A1cao (6) | cao; allow fraction or decimal, also allow $\dfrac{3 + 2\sqrt{2}}{4}$ |
---\begin{enumerate}
\item The discrete random variable $X$ has probability distribution,
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 3 & 7 \\
\hline
$\mathrm { P } ( X = x )$ & $p$ & $r$ & $p$ & 0.3 & $r$ \\
\hline
\end{tabular}
\end{center}
where $p$ and $r$ are probabilities.\\
Given that $\mathrm { E } ( X ) = 1.95$\\
find the exact value of $\mathrm { E } ( \sqrt { X + 1 } )$ giving your answer in the form $a + b \sqrt { 2 }$ where $a$ and $b$ are rational.\\
(6)
\hfill \mbox{\textit{Edexcel FS1 AS 2024 Q3 [6]}}