| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find multiple parameters from system |
| Difficulty | Standard +0.8 This S2 question requires setting up and solving a system of two equations (normalization condition and expectation) involving piecewise integration with parameters in both the integrand and limits. While the integration itself is routine, coordinating multiple constraints with two unknowns and managing the piecewise structure elevates this above standard single-parameter pdf questions. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| VIIIV SIHI NI IIIYM ION OC | VIIV SIHI NI JIIIM I ION OC | VI4V SIHI NI JIIYM IONOO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_{-1}^{2} k(x^2+a)\,dx + \int_{2}^{3} 3k\,dx = 1\) | M1 | Writing or using this expression, ignore limits |
| \(\left[k\left(\frac{x^3}{3}+ax\right)\right]_{-1}^{2} + [3kx]_{2}^{3} = 1\) | dM1 | Attempting to integrate at least one \(x^n \to \frac{x^{n+1}}{n+1}\) and sight of correct limits |
| \(k\left(\frac{8}{3}+2a+\frac{1}{3}+a\right)+9k-6k=1\) | A1 | Correct equation, need not be simplified |
| \(6k + 3ak = 1\) | ||
| \(\int_{-1}^{2} k(x^3+ax)\,dx + \int_{2}^{3} 3kx\,dx = \frac{17}{12}\) | M1 | Setting \(= \frac{17}{12}\), ignore limits |
| \(\left[k\left(\frac{x^4}{4}+\frac{ax^2}{2}\right)\right]_{-1}^{2} + \left[\frac{3kx^2}{2}\right]_{2}^{3} = \frac{17}{12}\) | dM1 | Attempting to integrate at least one \(x^n \to \frac{x^{n+1}}{n+1}\) and sight of correct limits |
| \(k\left(4+2a-\frac{1}{4}-\frac{a}{2}\right)+\frac{27k}{2}-6k=\frac{17}{12}\) | A1 | Correct equation, need not be simplified |
| \(\frac{45k}{4}+\frac{3ak}{2}=\frac{17}{12}\) | ||
| \(135k+18ak=17\), \(99k=11\) | ddM1 | Attempting to solve two simultaneous equations in \(a\) and \(k\) by eliminating one variable (dependent on 1st and 3rd M1s) |
| \(a=1,\; k=\frac{1}{9}\) | A1 | Both \(a\) and \(k\) correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\) | B1 |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_{-1}^{2} k(x^2+a)\,dx + \int_{2}^{3} 3k\,dx = 1$ | M1 | Writing or using this expression, ignore limits |
| $\left[k\left(\frac{x^3}{3}+ax\right)\right]_{-1}^{2} + [3kx]_{2}^{3} = 1$ | dM1 | Attempting to integrate at least one $x^n \to \frac{x^{n+1}}{n+1}$ and sight of correct limits |
| $k\left(\frac{8}{3}+2a+\frac{1}{3}+a\right)+9k-6k=1$ | A1 | Correct equation, need not be simplified |
| $6k + 3ak = 1$ | | |
| $\int_{-1}^{2} k(x^3+ax)\,dx + \int_{2}^{3} 3kx\,dx = \frac{17}{12}$ | M1 | Setting $= \frac{17}{12}$, ignore limits |
| $\left[k\left(\frac{x^4}{4}+\frac{ax^2}{2}\right)\right]_{-1}^{2} + \left[\frac{3kx^2}{2}\right]_{2}^{3} = \frac{17}{12}$ | dM1 | Attempting to integrate at least one $x^n \to \frac{x^{n+1}}{n+1}$ and sight of correct limits |
| $k\left(4+2a-\frac{1}{4}-\frac{a}{2}\right)+\frac{27k}{2}-6k=\frac{17}{12}$ | A1 | Correct equation, need not be simplified |
| $\frac{45k}{4}+\frac{3ak}{2}=\frac{17}{12}$ | | |
| $135k+18ak=17$, $99k=11$ | ddM1 | Attempting to solve two simultaneous equations in $a$ and $k$ by eliminating one variable (dependent on 1st and 3rd M1s) |
| $a=1,\; k=\frac{1}{9}$ | A1 | Both $a$ and $k$ correct |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2$ | B1 | |
---5. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by
$$f ( x ) = \left\{ \begin{array} { c c }
k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\
3 k & 2 < x \leqslant 3 \\
0 & \text { otherwise }
\end{array} \right.$$
where $k$ and $a$ are constants.\\
Given that $\mathrm { E } ( X ) = \frac { 17 } { 12 }$
\begin{enumerate}[label=(\alph*)]
\item find the value of $k$ and the value of $a$
\item Write down the mode of $X$
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIIIV SIHI NI IIIYM ION OC & VIIV SIHI NI JIIIM I ION OC & VI4V SIHI NI JIIYM IONOO \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q5 [9]}}