| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Expectation and variance from distribution |
| Difficulty | Standard +0.3 This is a standard Further Statistics 1 question requiring systematic application of probability axioms, expectation/variance formulas, and independence. Part (a) uses the probability sum constraint to express E(X) in terms of a and b, part (b) solves simultaneous equations using the variance condition, and part (c) applies independence to enumerate favorable outcomes. While it involves multiple steps and algebraic manipulation, the techniques are routine for FS1 students with no novel insight required, making it slightly easier than average A-level difficulty. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | - 2 | - 1 | 0 | 1 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.25 | \(a\) | \(b\) | \(a\) | 0.30 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \([E(X) =] -2 \times 0.25 + -1 \times a + 0 \times b + 1 \times a + 3 \times 0.3 = \mathbf{0.4}\) | M1, A1 | 1.1b, 1.1b |
| (b) \(E(X^2) = (-2)^2 \times 0.25 + (-1)^2 \times a + 0 \times 1^2 \times a + 3^2 \times 0.3 (= 2a + 3.7)\) | M1 | 2.1 |
| \([\text{Var}(X) =] 3.9 - 2a + 3.7 - ''0.4^2''\) \(a = \mathbf{0.18}\) | dM1, A1 | 1.1b, 1.1b |
| \([\text{Use of sum of probs} = 1 \text{ implies } 2a + b = 0.45]\) \(b = \mathbf{0.09}\) | A1 ft | 1.1b |
| (c) \(X_1 + X_2 > 3\) when \(X_1 = 3, X_2 = 1\) or \(X_1 = 1, X_2 = 3\) or \(X_1 = 3, X_2 = 3\) | M1 | 3.4 |
| \([\text{P}(X_1 + X_2 > 3) =]\) ''0.18'' \(\times 0.3 + 0.3 \times ''0.18'' + 0.3 \times 0.3\) or \(2 \times 0.3 \times (0.3 + ''0.18'') - 0.3^2 = \mathbf{0.198}\) | M1, A1 | 1.1b, 1.1b |
**(a)** $[E(X) =] -2 \times 0.25 + -1 \times a + 0 \times b + 1 \times a + 3 \times 0.3 = \mathbf{0.4}$ | M1, A1 | 1.1b, 1.1b | M1 for a correct attempt (at least 3 correct non-zero terms or products and addition) division by $k$ ($k \neq 1$) is M0; A1 for 0.4 o.e. (correct answer only scores 2 out of 2)
**(b)** $E(X^2) = (-2)^2 \times 0.25 + (-1)^2 \times a + 0 \times 1^2 \times a + 3^2 \times 0.3 (= 2a + 3.7)$ | M1 | 2.1
$[\text{Var}(X) =] 3.9 - 2a + 3.7 - ''0.4^2''$ $a = \mathbf{0.18}$ | dM1, A1 | 1.1b, 1.1b | 1st M1 for a correct attempt at $E(X^2)$ (at least 3 correct non-zero products and addition). Missing brackets around $-2$ and $-1$ is M0 unless recovered; 2nd dM1 (dep on 1st M1) for use of $3.9 = \text{their } E(X^2) - [E(X)]^2$ ft their $E(X) = 0.4$; 1st A1 for $a = 0.18$ o.e.
$[\text{Use of sum of probs} = 1 \text{ implies } 2a + b = 0.45]$ $b = \mathbf{0.09}$ | A1 ft | 1.1b | 2nd A1 (dep on 1st M1 only) for $b = 0.09$ o.e. or their $b = 0.45 - 2 \times ''a''$ (provided both $a$ and $b$ are probabilities) | (4)
**(c)** $X_1 + X_2 > 3$ when $X_1 = 3, X_2 = 1$ or $X_1 = 1, X_2 = 3$ or $X_1 = 3, X_2 = 3$ | M1 | 3.4 | 1st M1 for identifying at least 2 cases e.g. $X_1 = 3, X_2 > 1$ counts as 2 cases (ignore extras including any incorrect pairs identified) implied by at least two correct products of probs. or correct ft products of probs.
$[\text{P}(X_1 + X_2 > 3) =]$ ''0.18'' $\times 0.3 + 0.3 \times ''0.18'' + 0.3 \times 0.3$ or $2 \times 0.3 \times (0.3 + ''0.18'') - 0.3^2 = \mathbf{0.198}$ | M1, A1 | 1.1b, 1.1b | 2nd M1 for a correct numerical expression for the probability ft their ''0.18''; A1 for 0.198 o.e. | (3)
**(Total: 9 marks)**
---\begin{enumerate}
\item The discrete random variable $X$ has probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 2 & - 1 & 0 & 1 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & 0.25 & $a$ & $b$ & $a$ & 0.30 \\
\hline
\end{tabular}
\end{center}
where $a$ and $b$ are probabilities.\\
(a) Find $\mathrm { E } ( X )$
Given that $\operatorname { Var } ( X ) = 3.9$\\
(b) find the value of $a$ and the value of $b$
The independent random variables $X _ { 1 }$ and $X _ { 2 }$ each have the same distribution as $X$\\
(c) Find $\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 3 \right)$
\hfill \mbox{\textit{Edexcel FS1 2023 Q1 [9]}}