| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Variance of linear transformation |
| Difficulty | Moderate -0.8 This is a straightforward Further Statistics question testing basic properties of expectation and variance with discrete distributions. Part (a) uses E(aX)=aE(X), part (b) uses Var(aX)=a²Var(X), part (c) requires equating E and Var formulas (routine algebra), and part (d) is standard variance calculation from a probability distribution. All parts are direct applications of standard formulas with minimal problem-solving required. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(S\) | 2 | 3 | 4 | 5 | 6 |
| \(P ( S = S )\) | \(\frac { 2 } { 9 }\) | \(\frac { 1 } { 9 }\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 9 }\) | \(\frac { 2 } { 9 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(4\) | B1 [1] | Exact, allow 4.00 |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 \times \text{Var}(W)\) or \(\Sigma 4w^2 P(w) - 4^2\) | M1 | Can be implied by answer |
| \(\frac{8}{3}\) | A1 [2] | Can be written down, exact or awrt 2.67 |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 + k = \frac{8}{3}\) | M1 | Use *their* (a) \(+ k =\) *their* (b) |
| \(k = -\frac{4}{3}\) | A1 [2] | Exact or awrt \(-1.33\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma s^2 P(s)\ (= 18)\) | M1 | Clear attempt at finding \(\Sigma s^2 P(s)\) |
| \(18 - 4^2\) | M1 | Subtract \(4^2\) |
| \(2\) | A1 [3] | Exact, allow 2.00. Can scale variables, e.g. \(X = S - 3\): \(E(X^2) = \frac{18}{9} = 2\); \(E(X) = 0\); \(\text{Var}(S) = 2\) |
# Question 1:
## Part (a)
$4$ | B1 [1] | Exact, allow 4.00
## Part (b)
$4 \times \text{Var}(W)$ or $\Sigma 4w^2 P(w) - 4^2$ | M1 | Can be implied by answer
$\frac{8}{3}$ | A1 [2] | Can be written down, exact or awrt 2.67
## Part (c)
$4 + k = \frac{8}{3}$ | M1 | Use *their* **(a)** $+ k =$ *their* **(b)**
$k = -\frac{4}{3}$ | A1 [2] | Exact or awrt $-1.33$
## Part (d)
$\Sigma s^2 P(s)\ (= 18)$ | M1 | Clear attempt at finding $\Sigma s^2 P(s)$
$18 - 4^2$ | M1 | Subtract $4^2$
$2$ | A1 [3] | Exact, allow 2.00. Can scale variables, e.g. $X = S - 3$: $E(X^2) = \frac{18}{9} = 2$; $E(X) = 0$; $\text{Var}(S) = 2$
---1 The random variable $W$ can take values 1,2 or 3 and has a discrete uniform distribution.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $\mathrm { E } ( 2 W )$.
\item Find the value of $\operatorname { Var } ( 2 W )$.
\item Determine the value of the constant $k$ for which $\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )$.
The random variable $S$ has the probability distribution shown in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
$S$ & 2 & 3 & 4 & 5 & 6 \\
\hline
$P ( S = S )$ & $\frac { 2 } { 9 }$ & $\frac { 1 } { 9 }$ & $\frac { 1 } { 3 }$ & $\frac { 1 } { 9 }$ & $\frac { 2 } { 9 }$ \\
\hline
\end{tabular}
\end{center}
\item Calculate $\operatorname { Var } ( S )$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2024 Q1 [8]}}