| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard continuous uniform distribution formulas. Part (a) uses basic probability addition, (b) applies the variance formula Var(W) = (b-a)²/12 with simple algebra, and remaining parts use direct substitution. All techniques are routine textbook exercises requiring recall and mechanical application rather than problem-solving insight. |
| Spec | 5.02e Discrete uniform distribution5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(17 < W < k) = P(W < k) - P(W < 17) = \frac{53}{60}\left(1 - \frac{1}{5}\right) = \frac{1}{12}\) | M1 A1 | M1 for writing or using \(P(W < k) - P(W < 17)\) allow \(< \) or \(\leqslant\). Allow equivalent expressions e.g. \(P(W \geqslant 17) - P(W \geqslant k) = \frac{1}{5}\left(1 - \frac{53}{60}\right)\). A1 oe condone awrt 0.0833 condone \(\frac{1}{12}\) coming from \(\frac{13}{60}\) –1 or \(\left |
| (2) | ||
| (b)(i) \(\frac{(b - a)^2}{12} = 75\), \(\frac{b - 17}{b - a} = \frac{1}{5}\) or \(\frac{17 - a}{b - a} = \frac{4}{5}\) | B1, B1 | 1st B1 correct equation for variance. 2nd B1 either correct probability equation. Allow e.g. \(k\) in place of \((b - a)\) |
| \(\frac{(b - a)^2}{12} = 75 \Rightarrow (b - a) = 30\) | M1 | 1st M1 eliminating \((b - a)\) which must appear in both equations. |
| \(b = 23\) and \(a = -7\) | A1 | A1 both \(b = 23\) and \(a = -7\) correct answers imply all 4 marks |
| (4) | ||
| (b)(ii) \(P(W < k) = \frac{k - ("-7")}{23^" - ("-7")} = \frac{53}{60}\) or \(P(17 < W < k) = \frac{k - 17}{30} = \frac{1}{12}\) or \(P(W > k) = \frac{"23" - k}{23^" - ("-7")} = \frac{7}{60}\) | M1 | M1 probability expression using uniform distribution ft their values |
| \(k = 19.5\) | A1 | A1 \(k = 19.5\) oe cao |
| (2) | ||
| (c) \(P(-5 < W < 5) = \frac{5 - (-5)}{''23'' - ''("-7")''} = \frac{1}{3}\) | M1A1ft | M1 for 10/(their \(b\) – their \(a\)). A1ft \(\frac{1}{3}\) oe condone awrt 0.333 (Allow ff \(\frac{10}{\text{their}(b-a)}\) as exact fraction or evaluated to 3sf or better provided \(a < -5\) and \(b > 5\)) |
| (2) | ||
| (d) \(E(W^2) = \text{Var}(W) + (E(W))^2 = 75 + \left(\frac{''23'' + ''-7''}{2}\right)^2 = 139\) | M1 A1 | M1 use of \(E(W^2) = \text{Var}(W) + (E(W))^2\) with values substituted for Var(W) and E(W). ff their values of \(a\) and \(b\) allow any rearrangement. Must have a correct (ff) expression or value for E(W). Also allow \(\int_{-7}^{23} \frac{1}{30} w^2 dw\) |
| A1 139 cao | ||
| (2) | ||
| [Total 12] |
**(a)** $P(17 < W < k) = P(W < k) - P(W < 17) = \frac{53}{60}\left(1 - \frac{1}{5}\right) = \frac{1}{12}$ | M1 A1 | M1 for writing or using $P(W < k) - P(W < 17)$ allow $< $ or $\leqslant$. Allow equivalent expressions e.g. $P(W \geqslant 17) - P(W \geqslant k) = \frac{1}{5}\left(1 - \frac{53}{60}\right)$. A1 oe condone awrt 0.0833 condone $\frac{1}{12}$ coming from $\frac{13}{60}$ –1 or $\left|-\frac{1}{12}\right|$
| | (2)
**(b)(i)** $\frac{(b - a)^2}{12} = 75$, $\frac{b - 17}{b - a} = \frac{1}{5}$ or $\frac{17 - a}{b - a} = \frac{4}{5}$ | B1, B1 | 1st B1 correct equation for variance. 2nd B1 either correct probability equation. Allow e.g. $k$ in place of $(b - a)$
$\frac{(b - a)^2}{12} = 75 \Rightarrow (b - a) = 30$ | M1 | 1st M1 eliminating $(b - a)$ which must appear in both equations.
$b = 23$ and $a = -7$ | A1 | A1 both $b = 23$ and $a = -7$ correct answers imply all 4 marks
| | (4)
**(b)(ii)** $P(W < k) = \frac{k - ("-7")}{23^" - ("-7")} = \frac{53}{60}$ or $P(17 < W < k) = \frac{k - 17}{30} = \frac{1}{12}$ or $P(W > k) = \frac{"23" - k}{23^" - ("-7")} = \frac{7}{60}$ | M1 | M1 probability expression using uniform distribution ft their values
$k = 19.5$ | A1 | A1 $k = 19.5$ oe cao
| | (2)
**(c)** $P(-5 < W < 5) = \frac{5 - (-5)}{''23'' - ''("-7")''} = \frac{1}{3}$ | M1A1ft | M1 for 10/(their $b$ – their $a$). A1ft $\frac{1}{3}$ oe condone awrt 0.333 (Allow ff $\frac{10}{\text{their}(b-a)}$ as exact fraction or evaluated to 3sf or better provided $a < -5$ and $b > 5$)
| | (2)
**(d)** $E(W^2) = \text{Var}(W) + (E(W))^2 = 75 + \left(\frac{''23'' + ''-7''}{2}\right)^2 = 139$ | M1 A1 | M1 use of $E(W^2) = \text{Var}(W) + (E(W))^2$ with values substituted for Var(W) and E(W). ff their values of $a$ and $b$ allow any rearrangement. Must have a correct (ff) expression or value for E(W). Also allow $\int_{-7}^{23} \frac{1}{30} w^2 dw$
| | A1 139 cao
| | (2)
| | [Total 12]\begin{enumerate}
\item Akia selects at random a value from the continuous random variable $W$, which is uniformly distributed over the interval $[ a , b ]$
\end{enumerate}
The probability that Akia selects a value greater than 17 is $\frac { 1 } { 5 }$
The probability that Akia selects a value less than $k$ is $\frac { 53 } { 60 }$\\
(a) Find the probability that Akia selects a value between 17 and $k$
It is known that $\operatorname { Var } ( W ) = 75$\\
(b) (i) Find the value of $a$ and the value of $b$\\
(ii) Find the value of $k$\\
(c) Find $\mathrm { P } ( - 5 < W < 5 )$\\
(d) Find $\mathrm { E } \left( W ^ { 2 } \right)$
\hfill \mbox{\textit{Edexcel S2 2023 Q6 [12]}}