| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate and compare mean, median, mode |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring standard techniques: sketching a piecewise linear pdf, using the normalization condition ∫f(x)dx=1 to find c, calculating a probability by integration, and finding mean and variance using E(X) and E(X²). All steps are routine applications of AS Further Statistics formulas with symmetric triangular distribution making calculations simpler. Slightly easier than average due to the symmetry and algebraic simplicity. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (i) | A |
| Answer | Marks |
|---|---|
| [2] | At least one triangle |
| Answer | Marks |
|---|---|
| error in, c on y-axis) | Must have labels x, f(x) (or pdf on y- |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | B | ½ × 2c × c = 1 |
| So c = 1 | M1 |
| Answer | Marks |
|---|---|
| [2] | For integration must get to at least |
| AG | Allow |
| (ii) | P(X < ¼) = |
| Answer | Marks |
|---|---|
| 2 4 4 32 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| A1 | www | Allow 0.719 |
| (iii) | Mean = 0 |
| Answer | Marks |
|---|---|
| Standard deviation = 0.408 or or | B1 |
Question 3:
3 | (i) | A | B1
B1
[2] | At least one triangle
Fully correct including –c, c on x-
axis (do not penalise omission of, or
error in, c on y-axis) | Must have labels x, f(x) (or pdf on y-
axis)
(i) | B | ½ × 2c × c = 1
So c = 1 | M1
A1
[2] | For integration must get to at least
AG | Allow
(ii) | P(X < ¼) =
= 23/ or 0.71875
32
Or
P(X < ¼) =
= 23/ or 0.71875
32
Or
23
1133
2 4 4 32 | M1
A1
[2]
M1
A1
M1
A1 | www | Allow 0.719
(iii) | Mean = 0
E(X2) =
=
Var(X) =
Standard deviation = 0.408 or or | B1
M1
A1
A1FT
[4]3 The probability density function of the continuous random variable $X$ is given by
$$\mathrm { f } ( x ) = \begin{cases} c + x & - c \leqslant x \leqslant 0 \\ c - x & 0 \leqslant x \leqslant c \\ 0 & \text { otherwise } \end{cases}$$
where $c$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item (A) Sketch the graph of the probability density function.\\
(B) Show that $c = 1$.
\item Find $\mathrm { P } \left( X < \frac { 1 } { 4 } \right)$.
\item Find
\begin{itemize}
\item the mean of $X$,
\item the standard deviation of $X$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS 2018 Q3 [10]}}