| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Proof and derivation of E(X) and Var(X) |
| Difficulty | Moderate -0.5 This question involves straightforward application of expectation and variance formulas to a Bernoulli distribution, then using the sum property to derive binomial distribution parameters. While it requires understanding that Y is a sum of independent Bernoulli trials, the calculations are routine and the connection is explicitly guided. Easier than average due to the step-by-step structure and standard results. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (a) | E(X) = p |
| Answer | Marks |
|---|---|
| Var(π) = πβπ2 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (b) | π = π +π +β―+π |
| Answer | Marks |
|---|---|
| π | B1 |
| Answer | Marks |
|---|---|
| [6] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Seen |
Question 11:
11 | (a) | E(X) = p
E(π2) = π
Var(π) = πβπ2 | B1
M1
A1
[3] | 1.1
1.1
1.1
11 | (b) | π = π +π +β―+π
1 2 50
where p = 0.2,
giving E(π ) = 0.2, Var(π ) = 0.2Γ(1β0.2) = 0.16
π π
E(π) = E(π )+E(π )+β―+E(π )
1 2 50
= 50Γ E(π ) = 50Γ0.2 = 10
π
Var(π) = Var(π )+Var(π )+β―+Var(π )
1 2 50
= 50ΓVar(π ) = 50Γ0.16 = 8
π | B1
B1
M1
A1
M1
A1
[6] | 3.1a
1.1
2.1
1.1
2.1
1.1 | Seen
AG SCB1 for use of results of part a and E(Y) = 50ΓE(X),
(Where the Xi are all independent) . Not simply np.
AG SCB1 for use of results of part a and Var(Y) =
50ΓVar(X). Not simply npq.
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.11 The random variable $X$ takes the value 1 with probability $p$ and the value 0 with probability $1 - p$.
\begin{enumerate}[label=(\alph*)]
\item Find each of the following.
\begin{itemize}
\item $\mathrm { E } ( X )$
\item $\operatorname { Var } ( X )$
\item The random variable $Y \sim \mathrm {~B} ( 50,0.2 )$ has mean $\mu$ and variance $\sigma ^ { 2 }$.
\end{itemize}
Use the results of part (a) to prove that
\begin{itemize}
\item $\mu = 10$
\item $\sigma ^ { 2 } = 8$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2023 Q11 [9]}}