| Exam Board | Edexcel |
|---|---|
| Module | FS2 AS (Further Statistics 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Conditional probability with uniform |
| Difficulty | Standard +0.3 This is a straightforward continuous uniform distribution question requiring basic probability calculations and understanding of transformations. Part (a) is direct probability from uniform distribution, part (b) requires finding when both conditions hold (simple inequality solving), and part (c) needs comparison of E[R] vs E[2/R²]. While it involves transformations, the uniform distribution makes calculations simple and no complex integration or novel insight is required—slightly easier than average A-level. |
| Spec | 5.02e Discrete uniform distribution5.03g Cdf of transformed variables |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(T<1)=\frac{1-0.5}{2.5-0.5}=\frac{1}{4}\) | B1 | 0.25 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P\!\left(\{T<2.25\}\cap\left\{\frac{1}{T^2}<2.25\right\}\right)=P\!\left(\{T<2.25\}\cap\left\{T^2>\frac{4}{9}\right\}\right)\) | M1 | Determining the conditions for both numbers to be smaller than 2.25 |
\(P\!\left(\frac{2}{3}| M1 |
Use of uniform distribution for region of \(T\) |
|
| \(=\frac{19}{24}\) | A1 | allow awrt 0.792 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(R)=1.5\) | B1 | |
| \(E\!\left(\frac{2}{R^2}\right)=\int_{0.5}^{2.5}\left(\frac{1}{2.5-0.5}\right)\frac{2}{r^2}\,dr\) | M1 | Attempt to set up integral for Greta's expectation |
| \(\left[-\frac{1}{r}\right]_{0.5}^{2.5}\) | dM1 | dep on previous M1; integration of expectation |
| \(=1.6\) | A1 | |
| Greta is the expected winner since she has the higher expected value \((1.6>1.5)\) | A1 | Greta with correct supporting reason and all previous marks scored in (c) |
## Question 5:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(T<1)=\frac{1-0.5}{2.5-0.5}=\frac{1}{4}$ | B1 | 0.25 oe |
**(1 mark)**
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $P\!\left(\{T<2.25\}\cap\left\{\frac{1}{T^2}<2.25\right\}\right)=P\!\left(\{T<2.25\}\cap\left\{T^2>\frac{4}{9}\right\}\right)$ | M1 | Determining the conditions for both numbers to be smaller than 2.25 |
| $P\!\left(\frac{2}{3}<T<2.25\right)=\frac{2.25-\frac{2}{3}}{2.5-0.5}$ | M1 | Use of uniform distribution for region of $T$ |
| $=\frac{19}{24}$ | A1 | allow awrt 0.792 |
**(3 marks)**
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $E(R)=1.5$ | B1 | |
| $E\!\left(\frac{2}{R^2}\right)=\int_{0.5}^{2.5}\left(\frac{1}{2.5-0.5}\right)\frac{2}{r^2}\,dr$ | M1 | Attempt to set up integral for Greta's expectation |
| $\left[-\frac{1}{r}\right]_{0.5}^{2.5}$ | dM1 | dep on previous M1; integration of expectation |
| $=1.6$ | A1 | |
| Greta is the expected winner since she has the higher expected value $(1.6>1.5)$ | A1 | Greta with correct supporting reason and all previous marks scored in (c) |
**SC:** Use of $R=\frac{2}{R^2}\Rightarrow R=\sqrt[3]{2}\rightarrow 1.5>\sqrt[3]{2}(=1.25\ldots)$ therefore Raja is more likely to win a single game, scores B1M0M0A0A1.
**(5 marks)**\begin{enumerate}
\item The random variable $X$ has the continuous uniform distribution over the interval [0.5, 2.5]
\end{enumerate}
Talia selects a number, $T$, at random from the distribution of $X$\\
(a) Find $\mathrm { P } ( T < 1 )$
Malik takes Talia's number, $T$, and calculates his number, $M$, where $M = \frac { 1 } { T ^ { 2 } }$\\
(b) Find the probability that both $T$ and $M$ are less than 2.25
Raja and Greta play a game many times.\\
Each time they play they use a number, $R$, randomly selected from the distribution of $X$\\
Raja's score is $R$\\
Greta's score is $G$, where $G = \frac { 2 } { R ^ { 2 } }$\\
(c) Determine, giving a reason, who you would expect to have the higher total score.
\hfill \mbox{\textit{Edexcel FS2 AS 2022 Q5 [9]}}