| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.8 This is a straightforward discrete probability distribution question requiring only routine techniques: summing probabilities to find k, then calculating E(X) and Var(X) using standard formulas. The algebra is simple with only 4 values, and the methods are direct textbook applications with no problem-solving insight needed. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = 0.05\) | M1, A1 NB Answer given | 2 marks |
| (ii) \(E(X) = 1 \times 0.2 + 2 \times 0.3 + 3 \times 0.3 + 4 \times 0.2 = 2.5\) (or by inspection) | M1 for \(\sum rp\) (at least 3 terms correct), A1 CAO | |
| \(E(X^2) = 1 \times 0.2 + 4 \times 0.3 + 9 \times 0.3 + 16 \times 0.2 = 7.3\) | M1 for \(\sum r^2p\) (at least 3 terms correct), M1 dep for – their \(E(X)^2\) | |
| \(\text{Var}(X) = 7.3 - 2.5^2 = 1.05\) | A1 FT their \(E(X)\) provided Var\((X) > 0\) | 5 marks |
**(i)** $4k + 6k + 6k + 4k = 1$
$20k = 1$
$k = 0.05$ | M1, A1 NB Answer given | 2 marks
**(ii)** $E(X) = 1 \times 0.2 + 2 \times 0.3 + 3 \times 0.3 + 4 \times 0.2 = 2.5$ (or by inspection) | M1 for $\sum rp$ (at least 3 terms correct), A1 CAO |
$E(X^2) = 1 \times 0.2 + 4 \times 0.3 + 9 \times 0.3 + 16 \times 0.2 = 7.3$ | M1 for $\sum r^2p$ (at least 3 terms correct), M1 dep for – their $E(X)^2$ |
$\text{Var}(X) = 7.3 - 2.5^2 = 1.05$ | A1 FT their $E(X)$ provided Var$(X) > 0$ | 5 marks
**TOTAL: 7 marks**
---2 The probability distribution of the random variable $X$ is given by the formula
$$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4 .$$
(i) Show that $k = 0.05$.\\
(ii) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR MEI S1 2010 Q2 [7]}}