| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| 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 probability distribution question requiring basic summation (using ΣP(X=r)=1 to find k), then standard expectation and variance formulas. All steps are routine calculations with no conceptual challenges—easier than average A-level. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.04a Discrete probability distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum P(X=r) = 1 \Rightarrow k(7+12+15+16) = 1\) | M1 | Summing all probabilities = 1 |
| \(k(1\times7 + 2\times6 + 3\times5 + 4\times4) = k(7+12+15+16) = 50k = 1\) | A1 | |
| \(k = \frac{1}{50}\) | Shown |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \frac{1}{50}(1\times7 + 2\times12 + 3\times15 + 4\times16)\) | M1 | \(\sum r \cdot P(X=r)\) |
| \(= \frac{7+24+45+64}{50} = \frac{140}{50} = 2.8\) | A1 | |
| \(E(X^2) = \frac{1}{50}(1\times7 + 4\times12 + 9\times15 + 16\times16) = \frac{7+48+135+256}{50} = \frac{446}{50} = 8.92\) | M1 | \(\sum r^2 \cdot P(X=r)\) |
| \(Var(X) = 8.92 - 2.8^2 = 8.92 - 7.84 = 1.08\) | M1 A1 | \(E(X^2) - [E(X)]^2\) |
# Question 3:
## Part (i)
| $\sum P(X=r) = 1 \Rightarrow k(7+12+15+16) = 1$ | M1 | Summing all probabilities = 1 |
| $k(1\times7 + 2\times6 + 3\times5 + 4\times4) = k(7+12+15+16) = 50k = 1$ | A1 | |
| $k = \frac{1}{50}$ | | Shown |
## Part (ii)
| $E(X) = \frac{1}{50}(1\times7 + 2\times12 + 3\times15 + 4\times16)$ | M1 | $\sum r \cdot P(X=r)$ |
| $= \frac{7+24+45+64}{50} = \frac{140}{50} = 2.8$ | A1 | |
| $E(X^2) = \frac{1}{50}(1\times7 + 4\times12 + 9\times15 + 16\times16) = \frac{7+48+135+256}{50} = \frac{446}{50} = 8.92$ | M1 | $\sum r^2 \cdot P(X=r)$ |
| $Var(X) = 8.92 - 2.8^2 = 8.92 - 7.84 = 1.08$ | M1 A1 | $E(X^2) - [E(X)]^2$ |
---3 The score, $X$, obtained on a given throw of a biased, four-faced die is given by the probability distribution
$$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
(i) Show that $k = \frac { 1 } { 50 }$.\\
(ii) Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR MEI S1 2006 Q3 [7]}}