| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | E(g(X)) and Var(g(X)) by integration |
| Difficulty | Moderate -0.8 This is a straightforward S2 question requiring basic uniform distribution knowledge (writing pdf, calculating probability) and one application of E(g(X)) using the formula E(πR²) = ∫πr²f(r)dr. All steps are routine with no conceptual challenges beyond standard textbook exercises. |
| Spec | 5.02e Discrete uniform distribution5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(r) = \begin{cases} \frac{1}{4} & 5 \leq r \leq 9 \\ 0 & \text{otherwise} \end{cases}\) | B1 | Allow \(r < 5\) and \(r > 9\) instead of 0 otherwise; allow \(<\) instead of \(\leq\); any letter may be used; must have \(f(r)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(7 < R < 10) = 2 \times \frac{1}{4} = \frac{1}{2}\) | B1 | oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([E(A) = E(\pi R^2)]\); \(E(R^2) = \text{Var}(R) + [E(R)]^2\) or \(\int_5^9 \frac{r^2}{4}\,dr\) | M1 | Using correct formula for \(E(R^2)\); may be in any order or written in words |
| \(E(R) = 7\), \(\text{Var}(R) = \frac{4}{3}\) or \(\left[\frac{r^3}{12}\right]_5^9\) | B1 | \(\text{Var}(R) = \frac{4}{3}\) or awrt 1.33 and \(E(R) = 7\); may be implied by correct answer |
| \(= 50\frac{1}{3}\) | A1 | Allow awrt 50.3 |
| \(E(A) = 50\frac{1}{3}\pi\) oe | A1 | Allow exact multiple of \(\pi\) e.g. \(50.\overline{3}\pi\) or awrt 158; do not allow \(50.3\pi\) |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(r) = \begin{cases} \frac{1}{4} & 5 \leq r \leq 9 \\ 0 & \text{otherwise} \end{cases}$ | B1 | Allow $r < 5$ and $r > 9$ instead of 0 otherwise; allow $<$ instead of $\leq$; any letter may be used; must have $f(r)$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(7 < R < 10) = 2 \times \frac{1}{4} = \frac{1}{2}$ | B1 | oe |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[E(A) = E(\pi R^2)]$; $E(R^2) = \text{Var}(R) + [E(R)]^2$ or $\int_5^9 \frac{r^2}{4}\,dr$ | M1 | Using correct formula for $E(R^2)$; may be in any order or written in words |
| $E(R) = 7$, $\text{Var}(R) = \frac{4}{3}$ or $\left[\frac{r^3}{12}\right]_5^9$ | B1 | $\text{Var}(R) = \frac{4}{3}$ or awrt 1.33 **and** $E(R) = 7$; may be implied by correct answer |
| $= 50\frac{1}{3}$ | A1 | Allow awrt 50.3 |
| $E(A) = 50\frac{1}{3}\pi$ oe | A1 | Allow exact multiple of $\pi$ e.g. $50.\overline{3}\pi$ or awrt 158; do not allow $50.3\pi$ |
---\begin{enumerate}
\item The random variable $R$ has a continuous uniform distribution over the interval [5,9]\\
(a) Specify fully the probability density function of $R$.\\
(b) Find $\mathrm { P } ( 7 < R < 10 )$
\end{enumerate}
The random variable $A$ is the area of a circle radius $R \mathrm {~cm}$.\\
(c) Find $\mathrm { E } ( \mathrm { A } )$\\
\hfill \mbox{\textit{Edexcel S2 2016 Q3 [6]}}