| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Calculate simple probabilities |
| Difficulty | Moderate -0.8 This is a straightforward application of continuous uniform distribution with clearly defined intervals. Parts (a)-(c) require only basic probability calculations and sketching a standard CDF. Part (d) involves independence but uses simple complementary probability reasoning. All techniques are standard S2 material with no novel problem-solving required. |
| Spec | 5.02e Discrete uniform distribution5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{19}{24}\) | B1 | allow awrt 0.792 |
| (b) \(P( | R | > 3.5) = \frac{-3.5 - (-5)}{19 - (-5)} = \frac{19 - 3.5}{19 - (-5)} = \frac{17}{24}\) |
| Answer | Marks | Guidance |
|---|---|---|
| (c) M1 straight line with increasing gradient. Allow a horizontal line to the right of 19 and/or a horizontal line to the left of −5 | M1, A1 | A1 starting at (−5, 0) and finishing at (19, 1) Need to be clear labels for −5, 19 and 1. 0 may be labelled or implied on the x- axis |
| (d)(i) 1st M1 for P(R > 10) eg \(1 - \frac{19 - (-5)}{19 - (-5)}\) no need to simplify. Implied by 0.375 or \(\frac{27}{512}\) You may use their denominator from (a) | M1 | |
| 2nd M1 ["their P(R > 10)"]³ They may use their denominator from (a) otherwise ft their P(R > 10) only if it is clearly labelled. | M1 | |
| A1 allow awrt 0.0527 | ||
| (ii) M1 Use of \(1 - p^3\) (\(0 < p < 1\)) (none are greater than 10cm from origin) or \(3p^2(1 - p) + 3p(1 - p)^2 + (1 - p)\) (\(0 < p < 1\)) working needs to be shown | M1 | |
| A1 allow awrt 0.756 | A1 | |
| SC M1A0 for finding the P (exactly 1 is > 10cm) = \(\frac{225}{512}\) (= 0.439...) |
**(a)** $\frac{19}{24}$ | B1 | allow awrt 0.792
**(b)** $P(|R| > 3.5) = \frac{-3.5 - (-5)}{19 - (-5)} = \frac{19 - 3.5}{19 - (-5)} = \frac{17}{24}$ | M1, A1 | M1 sum of two regions from uniform distribution or $1 - \frac{3.5 - (-3.5)}{19 - (-5)} = 1 - \frac{7}{24}$ oe You may ft their denominator from (a). A1 allow awrt 0.708
SC M1A0 for P(−3.5 < R < 3.5) = $\frac{7}{24}$ (awrt 0.292) or for finding P(R > 3.5) = $\frac{31}{48}$ (awrt 0.646) and P(R < −3.5) = $\frac{1}{16}$ (0.0625)
**(c)** M1 straight line with increasing gradient. Allow a horizontal line to the right of 19 and/or a horizontal line to the left of −5 | M1, A1 | A1 starting at (−5, 0) and finishing at (19, 1) Need to be clear labels for −5, 19 and 1. 0 may be labelled or implied on the x- axis
**(d)(i)** 1st M1 for P(R > 10) eg $1 - \frac{19 - (-5)}{19 - (-5)}$ no need to simplify. Implied by 0.375 or $\frac{27}{512}$ You may use their denominator from (a) | M1 |
2nd M1 ["their P(R > 10)"]³ They may use their denominator from (a) otherwise ft their P(R > 10) only if it is clearly labelled. | M1 |
A1 allow awrt 0.0527 |
**(ii)** M1 Use of $1 - p^3$ ($0 < p < 1$) (none are greater than 10cm from origin) or $3p^2(1 - p) + 3p(1 - p)^2 + (1 - p)$ ($0 < p < 1$) working needs to be shown | M1 |
A1 allow awrt 0.756 | A1 |
SC M1A0 for finding the P (exactly 1 is > 10cm) = $\frac{225}{512}$ (= 0.439...) |
**[10 marks]**
---\begin{enumerate}
\item A point is to be randomly plotted on the $x$-axis, where the units are measured in cm .
\end{enumerate}
The random variable $R$ represents the $x$ coordinate of the point on the $x$-axis and $R$ is uniformly distributed over the interval [-5,19]
A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin.\\
(a) Find the exact probability that the point is plotted to the right of the origin.\\
(b) Find the exact probability that the point is plotted more than 3.5 cm away from the origin.\\
(c) Sketch the cumulative distribution function of $R$
Three independent points with $x$ coordinates $R _ { 1 } , R _ { 2 }$ and $R _ { 3 }$ are plotted on the $x$-axis.\\
(d) Find the exact probability that\\
(i) all three points are more than 10 cm from the origin\\
(ii) the point furthest from the origin is more than 10 cm from the origin.
\hfill \mbox{\textit{Edexcel S2 2022 Q3 [10]}}