| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics A AS (Further Statistics A AS) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Easy -1.2 This is a straightforward probability distribution question requiring only routine calculations: substituting values into a given formula, summing probabilities to find k, and applying standard expectation/variance formulas. All steps are mechanical with no problem-solving or novel insight required, making it easier than average A-level material. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(r\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = r )\) | \(4 k\) | \(10 k\) |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | r |
| [1] | 1.1 | |
| P(X = r) | 4k | 10k |
| (b) | 4k + 10k + 18k + 28k + 40k = 1 |
| Answer | Marks |
|---|---|
| (so k = 0.01) | B1 |
| Answer | Marks |
|---|---|
| [2] | 2.4 |
| 1.1 | For equation |
| Answer | Marks |
|---|---|
| (c) | 0.50 |
| Answer | Marks |
|---|---|
| r | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | For graph with axes correctly labelled |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | The distribution has negative skew | B1 |
| [1] | 1.1 | |
| (e) | E(X) = 3.9 | |
| Var(X) = 1.33 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | BC |
Question 1:
1 | (a) | r | 1 | 2 | 3 | 4 | 5 | B1
[1] | 1.1
P(X = r) | 4k | 10k | 18k | 28k | 40k
(b) | 4k + 10k + 18k + 28k + 40k = 1
100k = 1
(so k = 0.01) | B1
B1
[2] | 2.4
1.1 | For equation
For 100k = 1 leading to given answer AG
(c) | 0.50
0.40
)r
0.30
=X(P
0.20
0.10
0.00
1 2 3 4 5
r | B1
B1
[2] | 1.1
1.1 | For graph with axes correctly labelled
For line graph with correct heights
B0 if tops of lines/points joined.
(d) | The distribution has negative skew | B1
[1] | 1.1
(e) | E(X) = 3.9
Var(X) = 1.33 | B1
B1
[2] | 1.1a
1.1 | BC
BC1 The discrete random variable $X$ has probability distribution defined by
$$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } + 3 r \right) \text { for } r = 1,2,3,4,5 \text {, where } k \text { is a constant. }$$
\begin{enumerate}[label=(\alph*)]
\item Complete the table below, using the copy in the Printed Answer Booklet giving the probabilities in terms of $k$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$r$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$\mathrm { P } ( X = r )$ & $4 k$ & $10 k$ & & & \\
\hline
\end{tabular}
\end{center}
\item Show that the value of $k$ is 0.01 .
\item Draw a graph to illustrate the distribution.
\item Describe the shape of the distribution.
\item Find each of the following.
\begin{itemize}
\item $\mathrm { E } ( X )$
\item $\operatorname { Var } ( X )$
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2019 Q1 [8]}}