| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Geometric applications |
| Difficulty | Standard +0.8 This question requires students to work with a transformation of a uniform random variable (A to L where L = √A), apply the transformation to find probabilities using the CDF method, and calculate variance of the transformed variable. While the uniform distribution itself is straightforward, the square root transformation and variance calculation require careful application of probability theory beyond routine textbook exercises. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03g Cdf of transformed variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(L \geqslant 4.5) \Rightarrow P(A \geqslant 20.25)\) | ||
| \(P(A \geqslant 20.25) = (30 - 20.25) \times \dfrac{1}{20}\) | M1 | \((30 - 20.25) \times \dfrac{1}{20}\) |
| \(= 0.4875\) | A1 | cao (Allow \(0.488\) or \(\dfrac{39}{80}\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Var}(L) = E(L^2) - [E(L)]^2\) | ||
| \([E(L^2)] = E(A) = 20\) | B1 | For 20 |
| \(E(L) = E(\sqrt{A}) = \dfrac{1}{20}\int_{10}^{30} \sqrt{a}\, dA\) or \(E(L) = \dfrac{1}{10}\int_{\sqrt{10}}^{\sqrt{30}} L^2\, dL\) | M1 | Attempt to integrate \(\dfrac{1}{20}\int_{10}^{30}\sqrt{a}\,dA\) or \(\dfrac{1}{10}\int_{\sqrt{10}}^{\sqrt{30}}L^2\,dL\). Ignore limits, accept any letter |
| \(= \dfrac{1}{20}\left[\dfrac{2}{3}a^{\frac{3}{2}}\right]_{10}^{30}\) or \(\dfrac{1}{10}\left[\dfrac{l^3}{3}\right]_{\sqrt{10}}^{\sqrt{30}}\) | A1 | Fully correct integration. Accept any letter. Must have limits |
| \(= 4.4231\ldots\) | A1 | \(4.42\) or better |
| \(\text{Var}(L) = 20 - (4.4231\ldots)^2\) | M1 | Use of \(\text{Var}(L) = E(L^2) - [E(L)]^2\) ft their \(E(L^2)\) and \(E(L)\), provided \(\text{Var}(L) > 0\) |
| \(= 0.4358\ldots\) | A1 | awrt \(0.436\) |
# Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(L \geqslant 4.5) \Rightarrow P(A \geqslant 20.25)$ | | |
| $P(A \geqslant 20.25) = (30 - 20.25) \times \dfrac{1}{20}$ | M1 | $(30 - 20.25) \times \dfrac{1}{20}$ |
| $= 0.4875$ | A1 | cao (Allow $0.488$ or $\dfrac{39}{80}$) |
# Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Var}(L) = E(L^2) - [E(L)]^2$ | | |
| $[E(L^2)] = E(A) = 20$ | B1 | For 20 |
| $E(L) = E(\sqrt{A}) = \dfrac{1}{20}\int_{10}^{30} \sqrt{a}\, dA$ **or** $E(L) = \dfrac{1}{10}\int_{\sqrt{10}}^{\sqrt{30}} L^2\, dL$ | M1 | Attempt to integrate $\dfrac{1}{20}\int_{10}^{30}\sqrt{a}\,dA$ or $\dfrac{1}{10}\int_{\sqrt{10}}^{\sqrt{30}}L^2\,dL$. Ignore limits, accept any letter |
| $= \dfrac{1}{20}\left[\dfrac{2}{3}a^{\frac{3}{2}}\right]_{10}^{30}$ **or** $\dfrac{1}{10}\left[\dfrac{l^3}{3}\right]_{\sqrt{10}}^{\sqrt{30}}$ | A1 | Fully correct integration. Accept any letter. Must have limits |
| $= 4.4231\ldots$ | A1 | $4.42$ or better |
| $\text{Var}(L) = 20 - (4.4231\ldots)^2$ | M1 | Use of $\text{Var}(L) = E(L^2) - [E(L)]^2$ ft their $E(L^2)$ and $E(L)$, provided $\text{Var}(L) > 0$ |
| $= 0.4358\ldots$ | A1 | awrt $0.436$ |7 The sides of a square are each of length $L \mathrm {~cm}$ and its area is $A \mathrm {~cm} ^ { 2 }$
Given that $A$ is uniformly distributed on the interval [10,30]
\begin{enumerate}[label=(\alph*)]
\item find $\mathrm { P } ( L \geqslant 4.5 )$
\item find $\operatorname { Var } ( L )$\\
\begin{center}
\end{center}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{a009b02e-4cd3-497b-a141-4630c653e20b-28_2655_1947_114_116}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2022 Q7 [8]}}