| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from probability function |
| Difficulty | Moderate -0.3 This is a standard S1 probability distribution question testing routine procedures: finding a normalizing constant by summing probabilities to 1, calculating expectation using the definition, and applying variance formulas including the linear transformation rule. The symmetry of the distribution simplifies E(X)=0 immediately, and all calculations are straightforward arithmetic with no conceptual challenges beyond applying memorized formulas. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks |
|---|---|
| (a) \(c(9 + 4 + 1 + 1 + 4 + 9) = 1\) therefore \(c = \frac{1}{28}\) | M1 A1 A1 |
| Answer | Marks |
|---|---|
| (i) \(E(X) = 0\) | B1 M1 A1 |
| (ii) \(E(X^2) = \frac{81 + 16 + 1 + 1 + 16 + 81}{28} = 7\) | B1 M1 A1 |
| Answer | Marks |
|---|---|
| (i) \(\text{Var}(X) = 7\) | B1 M1 A1 |
| (ii) \(\text{Var}(10 - 2X) = 4\text{Var}(X) = 28\) | B1 M1 A1 |
**(a)** $c(9 + 4 + 1 + 1 + 4 + 9) = 1$ therefore $c = \frac{1}{28}$ | M1 A1 A1 |
**(b)**
(i) $E(X) = 0$ | B1 M1 A1 |
(ii) $E(X^2) = \frac{81 + 16 + 1 + 1 + 16 + 81}{28} = 7$ | B1 M1 A1 |
**(c)**
(i) $\text{Var}(X) = 7$ | B1 M1 A1 |
(ii) $\text{Var}(10 - 2X) = 4\text{Var}(X) = 28$ | B1 M1 A1 |
**Total marks: 9**
---The discrete random variable $X$ has probability function
$$P(X = x) = \begin{cases} cx^2 & x = -3, -2, -1, 1, 2, 3 \\ 0 & \text{otherwise.} \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $c = \frac{1}{28}$. [3 marks]
\item Calculate \begin{enumerate}[label=(\roman*)] \item $E(X)$, \item $E(X^2)$. \end{enumerate} [3 marks]
\item Calculate \begin{enumerate}[label=(\roman*)] \item $\text{Var}(X)$, \item $\text{Var}(10 - 2X)$. \end{enumerate} [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q3 [9]}}