| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate E(X) from cumulative distribution |
| Difficulty | Moderate -0.8 This is a straightforward S1 question requiring students to convert a cumulative distribution function to a probability distribution, then apply the standard E(X) formula. It involves only basic arithmetic with three values and follows a routine procedure taught in all S1 courses with no conceptual challenges or problem-solving required. |
| Spec | 5.02a Discrete probability distributions: general |
| VIAV SIHI NI III IM IONOOC | VIIIV SIHI NI III IM I I N O O | VI4V SIHI NI III IM I ON OC |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \([F(6)=]\frac{45}{77}\) and \([F(7)=]\frac{60}{77}\) | M1 | For \(\frac{45}{77}\) and \(\frac{60}{77}\) seen. Allow awrt 0.58 and awrt 0.78. May be seen unsimplified. Implied by \(2^{\text{nd}}\) M1 or by seeing \(\frac{15}{77}\) |
| \([P(W=7)=F(7)-F(6)=]\frac{60}{77}-\frac{45}{77}\left[=\frac{15}{77}\right]\) and \([P(W=8)=F(8)-F(7)=]1-\frac{60}{77}\left[=\frac{17}{77}\right]\) | M1 | For \(\frac{60}{77}-\frac{45}{77}\) and \(1-\frac{60}{77}\). Allow awrt 0.195 or 0.20 and awrt 0.22 ft their \(F(6)\) and \(F(7)\) if working shown |
| \(E(W)=6\times\frac{45}{77}+7\times\frac{15}{77}+8\times\frac{17}{77}\) | M1 | Attempt to calculate \(E(W)\) with \(P(W=6)\) correct and correct method or value for at least one of \(P(W=7)\) or \(P(W=8)\) |
| \(=\frac{73}{11}\) or awrt \(\mathbf{6.64}\) | A1 | \(\frac{73}{11}\) oe or awrt 6.64 |
# Question 4:
| Working | Mark | Guidance |
|---------|------|----------|
| $[F(6)=]\frac{45}{77}$ **and** $[F(7)=]\frac{60}{77}$ | M1 | For $\frac{45}{77}$ **and** $\frac{60}{77}$ seen. Allow awrt 0.58 **and** awrt 0.78. May be seen unsimplified. Implied by $2^{\text{nd}}$ M1 or by seeing $\frac{15}{77}$ |
| $[P(W=7)=F(7)-F(6)=]\frac{60}{77}-\frac{45}{77}\left[=\frac{15}{77}\right]$ **and** $[P(W=8)=F(8)-F(7)=]1-\frac{60}{77}\left[=\frac{17}{77}\right]$ | M1 | For $\frac{60}{77}-\frac{45}{77}$ **and** $1-\frac{60}{77}$. Allow awrt 0.195 or 0.20 **and** awrt 0.22 ft their $F(6)$ and $F(7)$ if working shown |
| $E(W)=6\times\frac{45}{77}+7\times\frac{15}{77}+8\times\frac{17}{77}$ | M1 | Attempt to calculate $E(W)$ with $P(W=6)$ correct and correct method or value for at least one of $P(W=7)$ or $P(W=8)$ |
| $=\frac{73}{11}$ or awrt $\mathbf{6.64}$ | A1 | $\frac{73}{11}$ oe or awrt 6.64 |\begin{enumerate}
\item The cumulative distribution function of the discrete random variable $W$, which takes only the values 6,7 and 8 , is given by
\end{enumerate}
$$F ( W ) = \frac { ( w + 3 ) ( w - 1 ) } { 77 } \text { for } w = 6,7,8$$
Find $\mathrm { E } ( W )$
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VIAV SIHI NI III IM IONOOC & VIIIV SIHI NI III IM I I N O O & VI4V SIHI NI III IM I ON OC \\
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\hfill \mbox{\textit{Edexcel S1 2022 Q4 [4]}}