| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(aX+b) transformations |
| Difficulty | Moderate -0.3 This is a straightforward multi-step calculation requiring finding the missing probability (a=0.4), computing E(D) and E(D²) from the distribution, then applying the standard variance transformation formula Var(aX+b)=a²Var(X). While it involves several calculations, each step uses routine techniques taught in Further Statistics with no conceptual challenges or novel insights required. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(d\) | 0 | 2 | 4 | 6 |
| \(\mathrm { P } ( D = d )\) | \(a\) | 0.1 | 0.3 | 0.2 |
| Answer | Marks |
|---|---|
| 2 | E(D) = 2×0.1 + 4×0.3 + 6×0.2 |
| Answer | Marks |
|---|---|
| = 50.76 | M1 |
| Answer | Marks |
|---|---|
| [7] | 2.1 |
| Answer | Marks |
|---|---|
| 3.4 | NB: a is not needed by this method |
| Answer | Marks |
|---|---|
| M1(use this formula), A1M1A1 | Or change |
Question 2:
2 | E(D) = 2×0.1 + 4×0.3 + 6×0.2
= 2.6
E(D2) = 22×0.1 + 42×0.3 + 62×0.2 [= 12.4]
Var(D) = 12.4 – 2.62
= 5.64
Var(3D + 4) = 9×Var(D)
= 50.76 | M1
A1
M1
M1
A1
M1
A1
[7] | 2.1
1.1
1.1
1.1
1.1
3.1a
3.4 | NB: a is not needed by this method
Or Σ(x – µ)2p(x)
Σp2d oe gets max M1A1M0M1M1
Allow even if their Var(D) < 0
SC: Σ(x – µ)2p(x): M1A1, a = 0.4 M1
M1(use this formula), A1M1A1 | Or change
0, 2, 4, 6 to
4, 10, 16, 22
and find a2 A discrete random variable $D$ has the following probability distribution, where $a$ is a constant.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$d$ & 0 & 2 & 4 & 6 \\
\hline
$\mathrm { P } ( D = d )$ & $a$ & 0.1 & 0.3 & 0.2 \\
\hline
\end{tabular}
\end{center}
Determine the value of $\operatorname { Var } ( 3 D + 4 )$.
\hfill \mbox{\textit{OCR Further Statistics 2021 Q2 [7]}}