| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find expectation E(X) |
| Difficulty | Standard +0.3 This is a standard S2 continuous probability distribution question requiring integration to find probabilities, verify a constant, and calculate E(X), Var(X), and E(g(X)). While it involves multiple parts and the function of a random variable in part (d) requires expanding and using linearity of expectation, these are routine techniques for S2. The geometric context (semicircle pdf) makes integration straightforward, and all steps follow standard procedures without requiring novel insight. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03d E(g(X)): general expectation formula |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{9}{20}\) | B1 | 0.45 oe cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((21k - k) \times \frac{\pi}{20} = 1\) | M1 | Use of the area of the rectangle \(= 1\). Any equivalent rearrangement, allow \(20k\) instead of \((21k-k)\) |
| \(k = \frac{1}{\pi}\) | A1* | Answer is given so a fully correct solution must be seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = \frac{1}{2}(k + 21k) = \frac{11}{\pi}\) | B1 | oe must be in terms of \(\pi\) (isw after correct answer seen) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Var}(X) = \frac{1}{12}(21k-k)^2\) or \(\text{Var}(X) = \int_{\frac{1}{\pi}}^{\frac{21}{\pi}} \frac{\pi}{20} x^2\, dx - \left(\frac{11}{\pi}\right)^2\) | M1 | Use of \(\frac{(b-a)^2}{12}\) or \(\text{Var}(X) = \int \frac{\pi}{20} x^2\, dx - \left(\frac{11}{\pi}\right)^2\) |
| \(= \frac{100}{3\pi^2}\) | A1 | oe must be in terms of \(\pi\) (isw after correct answer seen). SC: If both final answers given in terms of \(k\), score B1M1A0 for (c)(i) \(11k\) and (c)(ii) \(\frac{100}{3}k^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(A) = \pi E(X^2) + 4E(X) + \frac{4}{\pi}\) | M1 | For expanding \(E(A) = E\!\left(\pi X^2 + 4X + \frac{4}{\pi}\right)\) or for setting up correct integral (ignore limits) |
| \(E(X^2) = \frac{100}{3\pi^2} + \left(\frac{11}{\pi}\right)^2 = \frac{463}{3\pi^2}\) | M1 | Valid method for finding \(E(X^2)\) i.e. use of \(\text{Var}(X) + [E(X)]^2\) or integration of \(x^2 f(x)\) or integration of their \(f(x)A\) with at least one \(x^n \to x^{n+1}\) |
| \(E(A) = \frac{463}{3\pi} + \frac{44}{\pi} + \frac{4}{\pi} = \frac{607}{3\pi}\) | M1, A1 | Substitution of their \(E(X)\) and \(E(X^2)\) into their \(E(A)\) or use of correct limits; for \(\frac{607}{3\pi}\) or awrt 64.4 |
# Question 4:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{9}{20}$ | B1 | 0.45 oe cao |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(21k - k) \times \frac{\pi}{20} = 1$ | M1 | Use of the area of the rectangle $= 1$. Any equivalent rearrangement, allow $20k$ instead of $(21k-k)$ |
| $k = \frac{1}{\pi}$ | A1* | Answer is given so a fully correct solution must be seen |
## Part (c)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = \frac{1}{2}(k + 21k) = \frac{11}{\pi}$ | B1 | oe must be in terms of $\pi$ (isw after correct answer seen) |
## Part (c)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Var}(X) = \frac{1}{12}(21k-k)^2$ or $\text{Var}(X) = \int_{\frac{1}{\pi}}^{\frac{21}{\pi}} \frac{\pi}{20} x^2\, dx - \left(\frac{11}{\pi}\right)^2$ | M1 | Use of $\frac{(b-a)^2}{12}$ or $\text{Var}(X) = \int \frac{\pi}{20} x^2\, dx - \left(\frac{11}{\pi}\right)^2$ |
| $= \frac{100}{3\pi^2}$ | A1 | oe must be in terms of $\pi$ (isw after correct answer seen). SC: If both final answers given in terms of $k$, score B1M1A0 for (c)(i) $11k$ and (c)(ii) $\frac{100}{3}k^2$ |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(A) = \pi E(X^2) + 4E(X) + \frac{4}{\pi}$ | M1 | For expanding $E(A) = E\!\left(\pi X^2 + 4X + \frac{4}{\pi}\right)$ or for setting up correct integral (ignore limits) |
| $E(X^2) = \frac{100}{3\pi^2} + \left(\frac{11}{\pi}\right)^2 = \frac{463}{3\pi^2}$ | M1 | Valid method for finding $E(X^2)$ i.e. use of $\text{Var}(X) + [E(X)]^2$ or integration of $x^2 f(x)$ or integration of their $f(x)A$ with at least one $x^n \to x^{n+1}$ |
| $E(A) = \frac{463}{3\pi} + \frac{44}{\pi} + \frac{4}{\pi} = \frac{607}{3\pi}$ | M1, A1 | Substitution of their $E(X)$ and $E(X^2)$ into their $E(A)$ or use of correct limits; for $\frac{607}{3\pi}$ or awrt 64.4 |
---\begin{enumerate}
\item The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$, shown in the diagram, where $k$ is a constant.\\
\includegraphics[max width=\textwidth, alt={}, center]{f4fa6add-5860-4c88-bb70-f3edd9b22211-12_511_1096_351_351}\\
(a) Find $\mathrm { P } ( X < 10 k )$\\
(b) Show that $k = \frac { 1 } { \pi }$\\
(c) Find, in terms of $\pi$, the values of\\
(i) $\mathrm { E } ( X )$\\
(ii) $\operatorname { Var } ( X )$
\end{enumerate}
Circles are drawn with area $A$, where
$$A = \pi \left( X + \frac { 2 } { \pi } \right) ^ { 2 }$$
(d) Find $\mathrm { E } ( A )$
\hfill \mbox{\textit{Edexcel S2 2023 Q4 [10]}}