| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from table |
| Difficulty | Easy -1.8 This is a straightforward textbook exercise testing basic recall of expectation and variance formulas for discrete distributions. Parts (a) and (b) involve direct calculation from a given probability table with no problem-solving required. Parts (c)(i) and (c)(ii) test standard results about linear transformations of random variables (E(aX+b) and Var(aX+b)), which are routine applications once the formulas are known. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(a\) | 1 | 2 | 5 | 10 | 20 |
| \(\mathrm { P } ( A = a )\) | 0.3 | 0.1 | 0.1 | 0.2 | 0.3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma aP(A=a)\) | M1 | Use \(\Sigma aP(A=a)\) |
| \(= 9\) | A1 | Exact only |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma a^2P(A=a) = 143.2\) | M1 | Use \(\Sigma a^2P(A=a)\) |
| \(- 9^2\) | M1 | Subtract their \(\mu^2\) |
| \(= 62.2\) | A1 | Cao, any exact form |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(2A + 50)\) | M1 | Consider \(E(2A+50)\) |
| \(= 2\times9 + 50 \quad [= 68]\) | A1ft | FT on their \(E(A)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Var}(2A+50) = 4\times62.2 \quad [= 248.8]\) | B1ft | FT on their \(\text{Var}(A)\) |
## Question 1:
### Part (a):
$\Sigma aP(A=a)$ | **M1** | Use $\Sigma aP(A=a)$
$= 9$ | **A1** | Exact only
[2]
### Part (b):
$\Sigma a^2P(A=a) = 143.2$ | **M1** | Use $\Sigma a^2P(A=a)$
$- 9^2$ | **M1** | Subtract their $\mu^2$
$= 62.2$ | **A1** | Cao, any exact form
[3]
### Part (c)(i):
$E(2A + 50)$ | **M1** | Consider $E(2A+50)$
$= 2\times9 + 50 \quad [= 68]$ | **A1ft** | FT on their $E(A)$
[2]
### Part (c)(ii):
$\text{Var}(2A+50) = 4\times62.2 \quad [= 248.8]$ | **B1ft** | FT on their $\text{Var}(A)$
[1]
---1 The discrete random variable $A$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$a$ & 1 & 2 & 5 & 10 & 20 \\
\hline
$\mathrm { P } ( A = a )$ & 0.3 & 0.1 & 0.1 & 0.2 & 0.3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\mathrm { E } ( A )$.
\item Determine the value of $\operatorname { Var } ( A )$.
\item The variable $A$ represents the value in pence of a coin chosen at random from a pile. Mia picks one coin at random from the pile. She then adds, from a different source, another coin of the same value as the one that she has chosen, and one 50p coin.
\begin{enumerate}[label=(\roman*)]
\item Find the mean of the value of the three coins.
\item Find the variance of the value of the three coins.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2021 Q1 [8]}}