| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring standard techniques: summing probabilities to find k, then calculating E(X) and Var(X) using definitions. The arithmetic involves simple products like r(r+1) for r=1 to 5, making it slightly easier than average but still requiring careful calculation and knowledge of variance formula. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2k+6k+12k+20k+30k = 1\), \(70k = 1\) | M1 | For five multiples of \(k\) (at least four correct); do not need to sum or \(=1\) for M1 |
| \(k = \frac{1}{70}\) | A1 NB ANSWER GIVEN | Condone omission of either \(70k=1\) or \(k=1/70\) but not both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(X) = 1\times\frac{2}{70}+2\times\frac{6}{70}+3\times\frac{12}{70}+4\times\frac{20}{70}+5\times\frac{30}{70} = 4\) | M1, A1 CAO | M1 for \(\Sigma rp\) (at least 3 terms correct); 280/70 scores M1A0 |
| \(E(X^2) = 1\times\frac{2}{70}+4\times\frac{6}{70}+9\times\frac{12}{70}+16\times\frac{20}{70}+25\times\frac{30}{70} = \frac{1204}{70} = 17.2\) | M1 | M1 for \(\Sigma r^2p\) (at least 3 terms correct) |
| \(\text{Var}(X) = 17.2 - 4^2 = 1.2\) | M1dep, A1 | M1dep for their \(E(X)^2\); A1 FT their \(E(X)\) but not an error in \(E(X^2)\) provided \(\text{Var}(X)>0\); SC2 for use of \(1/70\) for all probabilities leading to \(E(X)=3/14\) and \(\text{Var}(X)=145/196=0.74\) |
## Question 4:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2k+6k+12k+20k+30k = 1$, $70k = 1$ | M1 | For five multiples of $k$ (at least four correct); do not need to sum or $=1$ for M1 |
| $k = \frac{1}{70}$ | A1 NB **ANSWER GIVEN** | Condone omission of either $70k=1$ or $k=1/70$ but not both |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = 1\times\frac{2}{70}+2\times\frac{6}{70}+3\times\frac{12}{70}+4\times\frac{20}{70}+5\times\frac{30}{70} = 4$ | M1, A1 CAO | M1 for $\Sigma rp$ (at least 3 terms correct); 280/70 scores M1A0 |
| $E(X^2) = 1\times\frac{2}{70}+4\times\frac{6}{70}+9\times\frac{12}{70}+16\times\frac{20}{70}+25\times\frac{30}{70} = \frac{1204}{70} = 17.2$ | M1 | M1 for $\Sigma r^2p$ (at least 3 terms correct) |
| $\text{Var}(X) = 17.2 - 4^2 = 1.2$ | M1dep, A1 | M1dep for their $E(X)^2$; A1 FT their $E(X)$ but not an error in $E(X^2)$ provided $\text{Var}(X)>0$; SC2 for use of $1/70$ for all probabilities leading to $E(X)=3/14$ and $\text{Var}(X)=145/196=0.74$ |
---4 The probability distribution of the random variable $X$ is given by the formula
$$\mathrm { P } ( X = r ) = k r ( r + 1 ) \quad \text { for } r = 1,2,3,4,5 .$$
(i) Show that $k = \frac { 1 } { 70 }$.\\
(ii) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR MEI S1 Q4 [7]}}