| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Minor (Further Statistics Minor) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Variance of sum of independent values |
| Difficulty | Standard +0.3 This question tests standard uniform distribution formulas with straightforward application. Part (a) requires counting integers below the midpoint (routine probability calculation), while part (b) applies the variance formula for uniform distributions and the property that Var(sum) = sum of variances for independent variables. The algebraic manipulation to reach the required form is mechanical rather than insightful. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02e Discrete uniform distribution5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | Number of values of X is ๐โ100+1 or ๐โ99 soi |
| Answer | Marks |
|---|---|
| 2(๐โ99) | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | N.B. may see use of ๐ = 2๐ |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (b) | 1 |
| Answer | Marks |
|---|---|
| 6 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 2.1 | Accept Var(๐) = 1 ((๐โ100)2โ1) |
Question 7:
7 | (a) | Number of values of X is ๐โ100+1 or ๐โ99 soi
100+๐ ๐โ100
(Number < is)
2 2
๐โ100
Probability = oe ISW
2(๐โ99) | M1
M1
A1
[3] | 3.1a
3.1a
1.1 | N.B. may see use of ๐ = 2๐
1๐โ50
1 1
e.g. (1โ ) or 2
2 ๐โ99 ๐โ99
7 | (b) | 1
Var(๐) = ((๐โ99)2โ1)
12
Var of sum of 50 values = 50ร 1 ((๐โ๐กโ๐๐๐ 99)2โ1)
12
= 25 (๐2โ198๐+9800)
6 | M1
M1
A1
[3] | 3.1a
1.1
2.1 | Accept Var(๐) = 1 ((๐โ100)2โ1)
12
Their 99 must be a positive integer
25
๐ = or exact equivalent
6
PMT
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Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.7 The discrete random variable $X$ has a uniform distribution over the set of all integers between 100 and $n$ inclusive, where $n$ is a positive integer with $n > 100$.
\begin{enumerate}[label=(\alph*)]
\item Given that $n$ is even, determine $\mathrm { P } \left( \mathrm { X } < \frac { 100 + \mathrm { n } } { 2 } \right)$.
\item Determine the variance of the sum of 50 independent values of $X$, giving your answer in the form $\mathrm { a } \left( \mathrm { n } ^ { 2 } + \mathrm { bn } + \mathrm { c } \right)$, where $a , b$ and $c$ are constants.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Minor 2023 Q7 [6]}}