| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | January |
| 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 solving a system of two equations (using the pdf integration condition and the expectation formula) to find two unknown parameters, involving piecewise integration with polynomial and constant terms. While the integration itself is routine, setting up and solving the simultaneous equations with the expectation condition requires careful algebraic manipulation, making it moderately challenging but still within standard S2 scope. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_{-1}^{2} k(x^2+a)\,dx + \int_{2}^{3} 3k\,dx = 1\) | M1 | Writing or using the equation equal to 1; ignore limits |
| \(\left[k\left(\frac{x^3}{3}+ax\right)\right]_{-1}^{2} + \left[3kx\right]_{2}^{3} = 1\) | dM1 | Attempting to integrate at least one \(x^n \to \frac{x^{n+1}}{n+1}\) and sight of correct limits |
| \(k\left(\frac{8}{3}+2a+\frac{1}{3}+a\right)+9k-6k=1 \Rightarrow 6k+3ak=1\) | A1 | Correct equation – need not be simplified |
| \(\int_{-1}^{2} k(x^3+ax)\,dx + \int_{2}^{3} 3kx\,dx = \frac{17}{12}\) | 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 |
| \(k\left(4+2a-\frac{1}{4}-\frac{a}{2}\right)+\frac{27k}{2}-6k=\frac{17}{12} \Rightarrow \frac{45k}{4}+\frac{3ak}{2}=\frac{17}{12}\) i.e. \(135k+18ak=17\) | A1 | Correct equation – need not be simplified |
| Solving simultaneously: \(99k = 11\) | ddM1 | Attempting to solve two simultaneous equations in \(a\) and \(k\) by eliminating 1 variable |
| \(a=1,\, k=\frac{1}{9}\) | A1 | Both \(a\) and \(k\) correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\) | B1 | — |
## Question 5:
### Part (a):
| $\int_{-1}^{2} k(x^2+a)\,dx + \int_{2}^{3} 3k\,dx = 1$ | M1 | Writing or using the equation equal to 1; ignore limits |
| $\left[k\left(\frac{x^3}{3}+ax\right)\right]_{-1}^{2} + \left[3kx\right]_{2}^{3} = 1$ | dM1 | Attempting to integrate at least one $x^n \to \frac{x^{n+1}}{n+1}$ and sight of correct limits |
| $k\left(\frac{8}{3}+2a+\frac{1}{3}+a\right)+9k-6k=1 \Rightarrow 6k+3ak=1$ | A1 | Correct equation – need not be simplified |
| $\int_{-1}^{2} k(x^3+ax)\,dx + \int_{2}^{3} 3kx\,dx = \frac{17}{12}$ | 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 |
| $k\left(4+2a-\frac{1}{4}-\frac{a}{2}\right)+\frac{27k}{2}-6k=\frac{17}{12} \Rightarrow \frac{45k}{4}+\frac{3ak}{2}=\frac{17}{12}$ i.e. $135k+18ak=17$ | A1 | Correct equation – need not be simplified |
| Solving simultaneously: $99k = 11$ | ddM1 | Attempting to solve two simultaneous equations in $a$ and $k$ by eliminating 1 variable |
| $a=1,\, k=\frac{1}{9}$ | A1 | Both $a$ and $k$ correct |
### Part (b):
| $2$ | 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 $X$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2015 Q5 [9]}}