| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Continuous CDF with polynomial pieces |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard CDF/PDF concepts: reading probabilities from a CDF, finding quartiles by solving F(x)=0.25, differentiating to get the PDF, and computing expectation/variance for a uniform distribution. All parts follow routine procedures with no novel insight required, making it slightly easier than average but not trivial due to the algebraic manipulation and multiple parts. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(X < 0) = F(0)\) | M1 | |
| \(= \dfrac{1}{k+1}\) | A1 | 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((q_1 + 1)\times\dfrac{1}{k+1} = \dfrac{1}{4}\) | M1 | Alternative from a sketch |
| \(q_1 + 1 = \dfrac{1}{4}(k+1)\) | A1 | |
| \(q_1 = \dfrac{1}{4}(k+1) - 1\) | A1 | 3 marks total; OE |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(f(x) = \dfrac{d}{dx}(F(x))\) | M1 | use of |
| \(= \dfrac{1}{k+1}\times\dfrac{d}{dx}(x+1)\) | ||
| \(= \dfrac{1}{k+1}, \quad -1\leq x \leq k\) | A1 | 2 marks total; AG; \(\dfrac{1}{k+1}\) clearly deduced |
| \(= 0\) otherwise |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(k=11 \Rightarrow f(x) = \begin{cases}\dfrac{1}{12} & -1\leq x \leq 11 \\ 0 & \text{otherwise}\end{cases}\) | ||
| Sketch: horizontal line on \([-1, 11]\) | B1 | |
| at \(f = \dfrac{1}{12}\) | B1 | 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(X) = \dfrac{1}{2}(-1+11) = 5\) | B1 | |
| \(\text{Var}(X) = \dfrac{1}{12}(11--1)^2 = 12\) | B1 | 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(q_1 < X < E(X)) = P(2 < X < 5)\) | ||
| \(= (5-2)\times\dfrac{1}{12}\) | M1 | |
| \(= 0.25\) | A1 | 2 marks total; AG |
## Question 8:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(X < 0) = F(0)$ | M1 | |
| $= \dfrac{1}{k+1}$ | A1 | 2 marks total |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $(q_1 + 1)\times\dfrac{1}{k+1} = \dfrac{1}{4}$ | M1 | Alternative from a sketch |
| $q_1 + 1 = \dfrac{1}{4}(k+1)$ | A1 | |
| $q_1 = \dfrac{1}{4}(k+1) - 1$ | A1 | 3 marks total; OE |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $f(x) = \dfrac{d}{dx}(F(x))$ | M1 | use of |
| $= \dfrac{1}{k+1}\times\dfrac{d}{dx}(x+1)$ | | |
| $= \dfrac{1}{k+1}, \quad -1\leq x \leq k$ | A1 | 2 marks total; AG; $\dfrac{1}{k+1}$ clearly deduced |
| $= 0$ otherwise | | |
### Part (d)(i):
| Working | Mark | Guidance |
|---------|------|----------|
| $k=11 \Rightarrow f(x) = \begin{cases}\dfrac{1}{12} & -1\leq x \leq 11 \\ 0 & \text{otherwise}\end{cases}$ | | |
| Sketch: horizontal line on $[-1, 11]$ | B1 | |
| at $f = \dfrac{1}{12}$ | B1 | 2 marks total |
### Part (d)(ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $E(X) = \dfrac{1}{2}(-1+11) = 5$ | B1 | |
| $\text{Var}(X) = \dfrac{1}{12}(11--1)^2 = 12$ | B1 | 2 marks total |
### Part (d)(iii):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(q_1 < X < E(X)) = P(2 < X < 5)$ | | |
| $= (5-2)\times\dfrac{1}{12}$ | M1 | |
| $= 0.25$ | A1 | 2 marks total; AG |8 The continuous random variable $X$ has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < - 1 \\
\frac { x + 1 } { k + 1 } & - 1 \leqslant x \leqslant k \\
1 & x > k
\end{array} \right.$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $k$, an expression for $\mathrm { P } ( X < 0 )$.
\item Determine an expression, in terms of $k$, for the lower quartile, $q _ { 1 }$.
\item Show that the probability density function of $X$ is defined by
$$\mathrm { f } ( x ) = \left\{ \begin{array} { c c }
\frac { 1 } { k + 1 } & - 1 \leqslant x \leqslant k \\
0 & \text { otherwise }
\end{array} \right.$$
\item Given that $k = 11$ :
\begin{enumerate}[label=(\roman*)]
\item sketch the graph of f;
\item determine $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$;
\item show that $\mathrm { P } \left( q _ { 1 } < X < \mathrm { E } ( X ) \right) = 0.25$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2008 Q8 [13]}}