| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Expectation and variance from distribution |
| Difficulty | Standard +0.3 This is a straightforward Further Statistics 1 question requiring standard application of variance formulas from a given probability distribution. Part (a) uses the basic variance formula E(X²) - [E(X)]², while part (b) requires treating X² as a new random variable and computing its variance, which adds one layer of complexity but remains algorithmic. The distribution is fully specified with simple values, making calculations routine for Further Maths students. |
| 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 | 5 |
| \(\mathrm { P } ( X = x )\) | 0.2 | 0.1 | 0.2 | 0.25 | 0.25 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(E(X) = [-1 \times 0.2 + 0 + 1 \times 0.2] + 3 \times 0.25 + 5 \times 0.25 [= 2]\) | M1 |
| \(E(X^2) = (-1)^2 \times 0.2 + 0 + 1 \times 0.2 + 3^2 \times 0.25 + 5^2 \times 0.25 [= 8.9]\) | M1 | 1.1b |
| \([\text{Var}(X) = 8.9 - 4 =] \mathbf{4.9}\) | A1 | 1.1b |
| (b) | \(E(X^4) = (-1)^4 \times 0.2 + 0 + 1 \times 0.2 + 3^4 \times 0.25 + 5^4 \times 0.25 [= 176.9]\) | M1 |
| \(\text{Var}(X^2) = \text{"} 176.9 \text{"} - \text{"} 8.9 \text{"}^2 = 97.69\) awrt \(\mathbf{97.7}\) | M1, A1 | 1.1b, 1.1b |
(a) | $E(X) = [-1 \times 0.2 + 0 + 1 \times 0.2] + 3 \times 0.25 + 5 \times 0.25 [= 2]$ | M1 | 1.1b |
| $E(X^2) = (-1)^2 \times 0.2 + 0 + 1 \times 0.2 + 3^2 \times 0.25 + 5^2 \times 0.25 [= 8.9]$ | M1 | 1.1b |
| $[\text{Var}(X) = 8.9 - 4 =] \mathbf{4.9}$ | A1 | 1.1b |
(b) | $E(X^4) = (-1)^4 \times 0.2 + 0 + 1 \times 0.2 + 3^4 \times 0.25 + 5^4 \times 0.25 [= 176.9]$ | M1 | 2.1 |
| $\text{Var}(X^2) = \text{"} 176.9 \text{"} - \text{"} 8.9 \text{"}^2 = 97.69$ awrt $\mathbf{97.7}$ | M1, A1 | 1.1b, 1.1b |
**Notes:**
- (a) 1st M1 for an attempt at $E(X)$, at least the final 2 products seen or an answer of 2
- (a) 2nd M1 for attempting a correct expression for $E(X^2)$, at least 3 non-zero products seen
- (a) A1 for 4.9 or exact equivalent
- (b) 1st M1 for attempting a correct expression, at least 3 non-zero products seen (implied by 176.9)
- (b) 2nd M1 for a correct method, ft their 176.9 and their 8.9 but must be intending $E(X^4)$ and $E(X^2)$
- (b) A1 for 97.69 or awrt 97.7
---\begin{enumerate}
\item The discrete random variable $X$ has the following probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 3 & 5 \\
\hline
$\mathrm { P } ( X = x )$ & 0.2 & 0.1 & 0.2 & 0.25 & 0.25 \\
\hline
\end{tabular}
\end{center}
(a) Find $\operatorname { Var } ( X )$\\
(b) Find $\operatorname { Var } \left( X ^ { 2 } \right)$
\hfill \mbox{\textit{Edexcel FS1 2024 Q1 [6]}}