| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Cumulative distribution function |
| Difficulty | Moderate -0.8 This is a straightforward question on the continuous uniform distribution requiring only direct application of the given CDF formula, recognition of the uniform distribution's properties, and standard variance calculation. Part (a) is simple substitution, part (b) is sketching a rectangular pdf, and part (c) uses the standard uniform variance formula with given values—all routine procedures with no problem-solving or novel insight required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | 12 ( a + b ) − a |
| Answer | Marks |
|---|---|
| b − a | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | Allow P ( X 12 ( a + b ) ) = 12 since this is a uniform |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (b) | 1 |
| Answer | Marks |
|---|---|
| All correct | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1.1 | For differentiation of F(x) Can be implied by correct diagram | |
| 2 | (c) | Var(X) = 3 |
| [1] | 1.1 |
Question 2:
2 | (a) | 12 ( a + b ) − a
P ( X 12 ( a + b ) ) =
b − a
12 ( b − a )
= = 12
b − a | M1
A1
[2] | 1.1a
1.1 | Allow P ( X 12 ( a + b ) ) = 12 since this is a uniform
distribution for B2
2 | (b) | 1
axb
f(x)=b−a
0 otherwise
Rectangle in correct position
All correct | M1
M1
A1
[3] | 1.1
1.1
1.1 | For differentiation of F(x) Can be implied by correct diagram
2 | (c) | Var(X) = 3 | B1
[1] | 1.12 The continuous random variable $X$ has cumulative distribution function given by\\
$F ( x ) = \begin{cases} 0 & x < a , \\ \frac { x - a } { b - a } & a \leqslant x \leqslant b , \\ 1 & x > b , \end{cases}$\\
where $a$ and $b$ are constants with $0 < \mathrm { a } < \mathrm { b }$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } \left( \mathrm { X } < \frac { 1 } { 2 } ( \mathrm { a } + \mathrm { b } ) \right)$.
\item Sketch the graph of the probability density function of $X$.
\item Find the variance of $X$ when $a = 2$ and $b = 8$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS 2022 Q2 [6]}}