| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Calculate probabilities from CDF |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring standard CDF manipulations: (a) evaluating P(W ≥ 3.5) = 1 - F(3.5) by substitution, (b) applying binomial distribution with the probability from (a), and (c) using the variance formula Var(W) = E(W²) - [E(W)]² where E(W²) requires integrating w² times the pdf (derivative of F). All techniques are routine for S2 students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1 - F(3.5) = 1 - 0.97127\ldots\) | M1 | For writing or using \(1 - F(3.5)\). Implied by correct answer |
| \(= 0.028727\ldots\) awrt \(0.0287\) | A1 | awrt \(0.0287\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(W \sim B(30, \text{"0.0287"})\) | M1 | For writing or using \(B(30, \text{"0.0287"})\); allow \(n(\text{"their } 0.0287\text{"})^1(1 - \text{"their } 0.0287\text{"})^{29}\); ignore any number for \(n\) (allow their \(p\) to 2sf) |
| \(1 - P(W \leq 1) = 1 - \left((1-\text{"0.0287"})^{30} + {}^{30}C_1(\text{"0.0287"})^1(1-\text{"0.0287"})^{29}\right)\) oe | M1 | For \(1 - \left((1-\text{"0.0287"})^{30} + {}^{30}C_1(\text{"0.0287"})^1(1-\text{"0.0287"})^{29}\right)\); allow \({}^{30}C_{29}\) in any form |
| \(= 1 - 0.78748\ldots = 0.2125\ldots\) awrt \(0.213\) to awrt \(0.216\) | A1 | Allow answer in the range awrt \(0.213\) to awrt \(0.216\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{dF(w)}{dw} = \frac{1}{3}\left(1 - \frac{w^3}{64}\right)\) | M1 | Differentiating \(F(w)\): at least one term correct |
| \(E(W^2) = \int_0^4 \frac{1}{3}\left(w^2 - \frac{w^5}{64}\right)dw = \frac{1}{3}\left[\frac{w^3}{3} - \frac{w^6}{384}\right]_0^4\) | dM1 | (Dep on previous M1). Attempting to integrate expanded \(w^2 f(w)\). At least one \(w^n \to w^{n+1}\). Ignore limits for this M mark |
| \(= \frac{32}{9}\) | A1 | awrt \(3.56\) must come from correct algebraic integration (may be embedded) |
| \(\text{Var}(W) = \frac{32}{9} - 1.6^2\) | M1 | Use of correct formula with values substituted. Must see the subtraction of \(1.6^2\) |
| \(= \frac{224}{225}\) | A1 | Dependent upon 2nd M1; awrt \(0.996\). (A correct answer with no algebraic integration seen may score M1M0A0M1A0) |
# Question 2:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $1 - F(3.5) = 1 - 0.97127\ldots$ | M1 | For writing or using $1 - F(3.5)$. Implied by correct answer |
| $= 0.028727\ldots$ awrt $0.0287$ | A1 | awrt $0.0287$ |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $W \sim B(30, \text{"0.0287"})$ | M1 | For writing or using $B(30, \text{"0.0287"})$; allow $n(\text{"their } 0.0287\text{"})^1(1 - \text{"their } 0.0287\text{"})^{29}$; ignore any number for $n$ (allow their $p$ to 2sf) |
| $1 - P(W \leq 1) = 1 - \left((1-\text{"0.0287"})^{30} + {}^{30}C_1(\text{"0.0287"})^1(1-\text{"0.0287"})^{29}\right)$ oe | M1 | For $1 - \left((1-\text{"0.0287"})^{30} + {}^{30}C_1(\text{"0.0287"})^1(1-\text{"0.0287"})^{29}\right)$; allow ${}^{30}C_{29}$ in any form |
| $= 1 - 0.78748\ldots = 0.2125\ldots$ awrt $0.213$ to awrt $0.216$ | A1 | Allow answer in the range awrt $0.213$ to awrt $0.216$ |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dF(w)}{dw} = \frac{1}{3}\left(1 - \frac{w^3}{64}\right)$ | M1 | Differentiating $F(w)$: at least one term correct |
| $E(W^2) = \int_0^4 \frac{1}{3}\left(w^2 - \frac{w^5}{64}\right)dw = \frac{1}{3}\left[\frac{w^3}{3} - \frac{w^6}{384}\right]_0^4$ | dM1 | (Dep on previous M1). Attempting to integrate expanded $w^2 f(w)$. At least one $w^n \to w^{n+1}$. Ignore limits for this M mark |
| $= \frac{32}{9}$ | A1 | awrt $3.56$ must come from correct algebraic integration (may be embedded) |
| $\text{Var}(W) = \frac{32}{9} - 1.6^2$ | M1 | Use of correct formula with values substituted. Must see the subtraction of $1.6^2$ |
| $= \frac{224}{225}$ | A1 | Dependent upon 2nd M1; awrt $0.996$. (A correct answer with no algebraic integration seen may score M1M0A0M1A0) |2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable $W$ with cumulative distribution function
$$\mathrm { F } ( w ) = \left\{ \begin{array} { c c }
0 & w < 0 \\
\frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4 \\
1 & w > 4
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling.
A random sample of 30 novice tightrope artists is taken.
\item Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling.
Given $\mathrm { E } ( W ) = 1.6$
\item use algebraic integration to find $\operatorname { Var } ( W )$\\
DO NOT WRITEIN THIS AREA
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2021 Q2 [10]}}