| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Derive or identify E(aX+b) or Var(aX+b) formulas |
| Difficulty | Easy -1.2 This is a straightforward application of standard expectation and variance formulas (E(aX+b) = aE(X)+b, Var(aX+b) = a²Var(X)) that students learn early in S1. Parts (a)-(c) are direct substitution, and part (d) requires expanding the square and applying linearity of expectation—routine algebraic manipulation with no problem-solving insight needed. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(2E(X) + 3 = 2a + 3\) | A1 | |
| (b) \(2^2 \times \text{Var}(X) = 4b\) | M1 A1 | |
| (c) \(\text{Var}(X) = E(X^2) - [E(X)]^2\) | B1 | |
| \(b = E(X^2) - a^2\) | M1 | |
| \(E(X^2) = a^2 + b\) | A1 | |
| (d) \(E[(X+1)^2] = E(X^2 + 2X + 1) = E(X^2) + 2E(X) + 1\) | M1 A1 | |
| \(= a^2 + b + 2a + 1 = (a+1)^2 + b\) | M1 A1 | (10 marks) |
(a) $2E(X) + 3 = 2a + 3$ | A1 |
(b) $2^2 \times \text{Var}(X) = 4b$ | M1 A1 |
(c) $\text{Var}(X) = E(X^2) - [E(X)]^2$ | B1 |
$b = E(X^2) - a^2$ | M1 |
$E(X^2) = a^2 + b$ | A1 |
(d) $E[(X+1)^2] = E(X^2 + 2X + 1) = E(X^2) + 2E(X) + 1$ | M1 A1 |
$= a^2 + b + 2a + 1 = (a+1)^2 + b$ | M1 A1 | (10 marks)
---3. The random variable $X$ is such that
$$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$
Find expressions in terms of $a$ and $b$ for
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { E } ( 2 X + 3 )$,
\item $\quad \operatorname { Var } ( 2 X + 3 )$,
\item $\mathrm { E } \left( X ^ { 2 } \right)$.
\item Show that
$$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q3 [10]}}