| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Standard +0.8 This requires setting up two simultaneous equations (probabilities sum to 1, and expectation formula), solving for x and y, then calculating E(W²) to find Var(W), and finally applying the variance transformation rule Var(aW+b) = a²Var(W). It's a multi-step problem requiring careful algebraic manipulation and knowledge of variance properties, making it moderately harder than average but still within standard Further Stats techniques. |
| Spec | 5.02b Expectation and variance: discrete random variables |
| \(w\) | 0 | 1 | 2 | 3 |
| \(\mathrm{P}(W = w)\) | 0.19 | 0.18 | \(x\) | \(y\) |
| Answer | Marks |
|---|---|
| 2 | x(cid:14) y(cid:32)0.63 |
| Answer | Marks |
|---|---|
| Var(cid:11)3W (cid:14)2(cid:12)(cid:32)32Var(W)(cid:32)8.6211 | M1 |
| Answer | Marks |
|---|---|
| [7] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2a | m |
Question 2:
2 | x(cid:14) y(cid:32)0.63
0.18(cid:14)2x(cid:14)3y(cid:32)1.61
x(cid:32)0.46, y(cid:32)0.17
p
S
So Var(W)(cid:32)0.18(cid:14)22x(cid:14)32y
(cid:16)1.612
(cid:32)0.9579
Var(cid:11)3W (cid:14)2(cid:12)(cid:32)32Var(W)(cid:32)8.6211 | M1
c
e
A1
A1
M1
M1
A1
A1FT
[7] | 1.1
i
1.1
1.1
3.1a
1.1
1.1
2.2a | m
Use (cid:166)p(cid:32)1and(cid:166)xp(cid:32)1.61to
set up two equations in x and y
and attempt to solve
For both equations
Correct values of x, y, these or
exact equivalent BC
Attempt (cid:166)x2p
Subtract their (cid:80)
Answer, exact or 0.958 only (no
FT)
FT 9(cid:117)their Var(W)
2
n
e
m
i
c
e
p
SThe probability distribution of a discrete random variable $W$ is given in the table.
\begin{tabular}{|c|c|c|c|c|}
\hline
$w$ & 0 & 1 & 2 & 3 \\
\hline
$\mathrm{P}(W = w)$ & 0.19 & 0.18 & $x$ & $y$ \\
\hline
\end{tabular}
Given that $\mathrm{E}(W) = 1.61$, find the value of $\mathrm{Var}(3W + 2)$. [7]
\hfill \mbox{\textit{OCR Further Statistics AS Q2 [7]}}