| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Multi-part piecewise CDF |
| Difficulty | Standard +0.8 This S2 question requires understanding CDF properties (continuity, F(3)=1), differentiation to find the pdf, and finding the mode. While multi-step, it follows standard procedures: using continuity conditions gives two equations for a and b, then routine calculus. The piecewise nature and algebraic manipulation add moderate complexity beyond typical S2 questions, but no novel insight is required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(4a = a(b) \Rightarrow b = 4\) | B1*cso | Answer given, need to see \(4a = a(b)\); allow \(4a(1) = a(b(1)-1+1)\) followed by \(b=4\) |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a(27b - 81 + 1) = 1\) | M1 | For a correct equation |
| \(a = \frac{1}{28}\) | A1 | 1/28 o.e. |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X > 2.25) = 1 - F(2.25)\) | M1 | For \(1 - F(2.25)\) or \(F(3) - F(2.25)\). Implied by correct answer |
| \(= 0.25237\ldots\) | A1 | awrt 0.252 |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f(x) = \frac{3}{7}x^2 - \frac{1}{7}x^3\) or \(\frac{2}{7}x\) | M1 | Differentiating to find \(f(x)\), one term correct or correct follow through. Condone missing \(a\). Differentiation may be seen anywhere. \(f(x) = a(12x^2 - 4x^3)\) or \(8ax\) |
| Sketch: straight line followed by smooth curve with mode near middle | B1 | Sketch of pdf. Straight line followed by smooth curve with mode near middle. Must be connected (no gap). Values not required, must begin and end on horizontal axis |
| (part of 5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f'(x) = \frac{6}{7}x - \frac{3}{7}x^2\) | dM1 | Dep on 1st M. Differentiating \(f(x)\) for \(1 < x \leq 3\) to find \(f'(x)\). \(x^n \to x^{n-1}\). Condone missing \(a\). \(f'(x) = a(24x - 12x^2)\) |
| \(\frac{6}{7}x - \frac{3}{7}x^2 = 0\) | dM1 | Dep on previous M. Putting their \(f'(x) = 0\) |
| Mode \(= 2\) | A1 | All but the B1 mark must be awarded |
| (5) | ||
| Total 10 |
# Question 3:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $4a = a(b) \Rightarrow b = 4$ | B1*cso | Answer given, need to see $4a = a(b)$; allow $4a(1) = a(b(1)-1+1)$ followed by $b=4$ |
| | **(1)** | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $a(27b - 81 + 1) = 1$ | M1 | For a correct equation |
| $a = \frac{1}{28}$ | A1 | 1/28 o.e. |
| | **(2)** | |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X > 2.25) = 1 - F(2.25)$ | M1 | For $1 - F(2.25)$ or $F(3) - F(2.25)$. Implied by correct answer |
| $= 0.25237\ldots$ | A1 | awrt 0.252 |
| | **(2)** | |
## Part (d)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $f(x) = \frac{3}{7}x^2 - \frac{1}{7}x^3$ or $\frac{2}{7}x$ | M1 | Differentiating to find $f(x)$, one term correct or correct follow through. Condone missing $a$. Differentiation may be seen anywhere. $f(x) = a(12x^2 - 4x^3)$ or $8ax$ |
| Sketch: straight line followed by smooth curve with mode near middle | B1 | Sketch of pdf. Straight line followed by smooth curve with mode near middle. Must be connected (no gap). Values not required, must begin and end on horizontal axis |
| | **(part of 5)** | |
## Part (d)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $f'(x) = \frac{6}{7}x - \frac{3}{7}x^2$ | dM1 | Dep on 1st M. Differentiating $f(x)$ for $1 < x \leq 3$ to find $f'(x)$. $x^n \to x^{n-1}$. Condone missing $a$. $f'(x) = a(24x - 12x^2)$ |
| $\frac{6}{7}x - \frac{3}{7}x^2 = 0$ | dM1 | Dep on previous M. Putting their $f'(x) = 0$ |
| Mode $= 2$ | A1 | All but the B1 mark must be awarded |
| | **(5)** | |
| | **Total 10** | |
---3. A continuous random variable $X$ has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r }
0 & x < 0 \\
4 a x ^ { 2 } & 0 \leqslant x \leqslant 1 \\
a \left( b x ^ { 3 } - x ^ { 4 } + 1 \right) & 1 < x \leqslant 3 \\
1 & x > 3
\end{array} \right.$$
where $a$ and $b$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Show that $b = 4$
\item Find the exact value of $a$
\item Find $\mathrm { P } ( X > 2.25 )$
\item Showing your working clearly,
\begin{enumerate}[label=(\roman*)]
\item sketch the probability density function of $X$
\item calculate the mode of $X$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2021 Q3 [10]}}