| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Joint distribution with covariance calculation |
| Difficulty | Standard +0.3 This is a straightforward joint distribution question requiring routine calculations of marginal distributions, expectations, and covariance using standard formulas. Part (i) involves mechanical computation of E(XY) - E(X)E(Y), while part (ii) is direct application of conditional probability. No problem-solving insight needed, just careful arithmetic with a provided table. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| \(X\) | |||
| \cline { 2 - 5 } \multicolumn{1}{l}{} | - 1 | 0 | 1 |
| 1 | 0.24 | 0.22 | 0.04 |
| 2 | 0.26 | 0.18 | 0.06 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substantially correct network diagram with activities A, B, C, D, E, F, G and dummy activities | M1 | Condone arrows missing or wrong way round, no end and/or extra dummies. Do NOT allow activity on node formulation |
| Correct network with arrows on at least the dummy activities, no extra dummies and a single end point | A1 | |
| A dummy is needed after \(C\) because \(D\) follows both \(B\) and \(C\) | B1 | A valid explanation |
| A dummy is needed after \(D\) because \(F\) and \(G\) both follow \(D\) | B1 | A valid explanation |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substantially correct forward pass | M1 | |
| Early event times correct (ft their network if possible) | A1 | |
| Substantially correct backwards pass | M1 | |
| Late event times correct (ft their network if possible) | A1 | |
| Minimum completion time \(= 14\) days | B1 | For 14, cao |
| Critical activities are \(A, C, D, F\) | B1 | For these four activities and no others, cao. In both cases these need to be stated, not implied from the diagram |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Resource histogram showing number of workers for different activities, with scales and labels, some days with 4 workers | M1 | For a reasonable attempt at using the number of workers for the different activities |
| Reasonable attempt with no overhanging blocks | M1 dep | |
| Entirely correct histogram | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E\) cannot happen until after \(C\) has finished so must overlap with \(F\) | B1 | Earliest finish for \(E >\) latest start for \(F\) |
| Start \(E\) immediately after \(C\) but delay the start of \(F\) for 1 day (until after \(E\) has finished) | B1 | For delaying the start of \(F\) (by 1 day) |
| 2/15 |
# Question 3:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Substantially correct network diagram with activities A, B, C, D, E, F, G and dummy activities | M1 | Condone arrows missing or wrong way round, no end and/or extra dummies. Do NOT allow activity on node formulation |
| Correct network with arrows on at least the dummy activities, no extra dummies and a single end point | A1 | |
| A dummy is needed after $C$ because $D$ follows both $B$ and $C$ | B1 | A valid explanation |
| A dummy is needed after $D$ because $F$ and $G$ both follow $D$ | B1 | A valid explanation |
| | **4** | |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Substantially correct forward pass | M1 | |
| Early event times correct (ft their network if possible) | A1 | |
| Substantially correct backwards pass | M1 | |
| Late event times correct (ft their network if possible) | A1 | |
| Minimum completion time $= 14$ days | B1 | For 14, cao |
| Critical activities are $A, C, D, F$ | B1 | For these four activities and no others, cao. In both cases these need to be stated, not implied from the diagram |
| | **6** | |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Resource histogram showing number of workers for different activities, with scales and labels, some days with 4 workers | M1 | For a reasonable attempt at using the number of workers for the different activities |
| Reasonable attempt with no overhanging blocks | M1 dep | |
| Entirely correct histogram | A1 | |
| | **3** | |
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E$ cannot happen until after $C$ has finished so must overlap with $F$ | B1 | Earliest finish for $E >$ latest start for $F$ |
| Start $E$ immediately after $C$ but delay the start of $F$ for 1 day (until after $E$ has finished) | B1 | For delaying the start of $F$ (by 1 day) |
| | **2/15** | |
---3 The discrete random variables $X$ and $Y$ have the joint probability distribution given in the following table.
\begin{center}
\begin{tabular}{ c | c | c c c | }
\multicolumn{1}{c}{} & \multicolumn{2}{c}{$X$} & \\
\cline { 2 - 5 }
\multicolumn{1}{l}{} & - 1 & 0 & 1 \\
\hline
1 & 0.24 & 0.22 & 0.04 \\
2 & 0.26 & 0.18 & 0.06 \\
\hline
\end{tabular}
\end{center}
(i) Show that $\operatorname { Cov } ( X , Y ) = 0$.\\
(ii) Find the conditional distribution of $X$ given that $Y = 2$.
\hfill \mbox{\textit{OCR S4 2007 Q3 [7]}}