| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Minor (Further Statistics Minor) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Variance of sum of independent values |
| Difficulty | Standard +0.3 This is a straightforward discrete uniform distribution question requiring standard formulas for mean and variance, basic properties of sums of independent random variables, and enumeration of outcomes for a convolution. While part (iii) requires some careful counting of the most likely sum, all techniques are routine for Further Statistics students with no novel problem-solving required. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02e Discrete uniform distribution |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (i) | E(X) = 5 |
| Answer | Marks |
|---|---|
| (cid:32)10 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (ii) | For 2 visits mean = 10 |
| Variance = 10 + 10 = 20 | B1FT |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | N |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (iii) | [Working leading to] 121 possibilities |
| Answer | Marks |
|---|---|
| 121 | M1 |
| Answer | Marks |
|---|---|
| [4] | 3.1b |
| Answer | Marks |
|---|---|
| 3.2a | E |
Question 3:
3 | (i) | E(X) = 5
10(cid:117)12
Var (cid:11)X(cid:12) (cid:32)
12
(cid:32)10 | B1
M1
A1
[3] | 1.1
3.3
1.1
3 | (ii) | For 2 visits mean = 10
Variance = 10 + 10 = 20 | B1FT
M1
A1
[3] | 1.1
3.4
1.1 | N
FT their (i)
3 | (iii) | [Working leading to] 121 possibilities
Most likely total is 10.
E.g. This is the one on the main diagonal of the sample
space so occurs most often.
11
Probability = oe
121 | M1
I
A1
C
E1
B1FT
[4] | 3.1b
M
1.1
2.4
3.2a | E
E.g. starting list (0, 10), (0, 9)
or as sample space or
calculation 11 × 11. Implied
by probability out of 121
Justification
FT correct probability for their
most likely totalA website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between $0$ and $10$ (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.
\begin{enumerate}[label=(\roman*)]
\item Let $X$ be the number of points gained after shopping once.
Find
\begin{itemize}
\item the mean of $X$
\item the variance of $X$. [3]
\end{itemize}
\item Let $Y$ be the number of points gained after shopping twice.
Find
\begin{itemize}
\item the mean of $Y$
\item the variance of $Y$. [3]
\end{itemize}
\item Find the probability of the most likely number of points gained after shopping twice. Justify your answer. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Minor Q3 [10]}}