| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Pure expectation and variance calculation |
| Difficulty | Moderate -0.3 This is a straightforward application of standard results for linear combinations of independent random variables. Part (i) requires direct use of E(Y-X)=E(Y)-E(X) and Var(Y-X)=Var(Y)+Var(X) with given distributions. Part (ii) applies the normal approximation (both B(32,0.5) and Po(28) satisfy conditions). Part (iii) is routine probability calculation with continuity correction. All steps are textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.02d Binomial: mean np and variance np(1-p)5.02m Poisson: mean = variance = lambda5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X)=16,\ E(Y)=28,\ E(Y-X)=12\) | B1 | |
| \(Var(X)=8,\ Var(Y)=28,\ 28+32\times\frac{1}{2}\times\frac{1}{2}\) | M1 | |
| \(Var(Y-X) = 36\) | A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(X \approx N(16,8)\) since \(32\times\frac{1}{2}>5\) and \(32\times\frac{1}{2}\times\frac{1}{2}>5\) | M1 | from \(np>5\) and \(nq>5\) or \(npq>5\) |
| \(Y \approx N(28,28)\) since \(E(Y)=28\) is large (eg \(>15\)) | M1 | |
| \(Y - X \approx N(12,\ 36)\) | B1ft | N(parameters from (i)) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(Y-X \leq -2.5) + P(Y-X \geq 2.5)\) | M1 | No CC or wrong CC. M1 \(\pm3\) or \(\pm3.5\) instead of \(\pm2.5\ (-2.5,-1.5)\) or \((-2.583,-1.417)\). A1M1 as main scheme A0 |
| \(= \phi(-2.417) + 1 - \phi(-1.583)\) | A1 M1 | \((-2.417, -1.583)\). Correct use of \(z\) and \(\Phi\) for their values. Allow M1A0 if only 1 interval, except M1A1 if \(0.0079\) or \(0.9433\) seen (SC) |
| \(0.0079 + 0.9433 = 0.951(2)\) | A1 | |
| [4] |
# Question 5:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X)=16,\ E(Y)=28,\ E(Y-X)=12$ | B1 | |
| $Var(X)=8,\ Var(Y)=28,\ 28+32\times\frac{1}{2}\times\frac{1}{2}$ | M1 | |
| $Var(Y-X) = 36$ | A1 | |
| | **[3]** | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \approx N(16,8)$ since $32\times\frac{1}{2}>5$ and $32\times\frac{1}{2}\times\frac{1}{2}>5$ | M1 | from $np>5$ and $nq>5$ or $npq>5$ |
| $Y \approx N(28,28)$ since $E(Y)=28$ is large (eg $>15$) | M1 | |
| $Y - X \approx N(12,\ 36)$ | B1ft | N(parameters from (i)) |
| | **[3]** | |
## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(Y-X \leq -2.5) + P(Y-X \geq 2.5)$ | M1 | No CC or wrong CC. M1 $\pm3$ or $\pm3.5$ instead of $\pm2.5\ (-2.5,-1.5)$ or $(-2.583,-1.417)$. A1M1 as main scheme A0 |
| $= \phi(-2.417) + 1 - \phi(-1.583)$ | A1 M1 | $(-2.417, -1.583)$. Correct use of $z$ and $\Phi$ for their values. Allow M1A0 if only 1 interval, except M1A1 if $0.0079$ or $0.9433$ seen (SC) |
| $0.0079 + 0.9433 = 0.951(2)$ | A1 | |
| | **[4]** | |
---5 The discrete random variables $X$ and $Y$ are independent with $X \sim \mathrm {~B} \left( 32 , \frac { 1 } { 2 } \right)$ and $Y \sim \operatorname { Po } ( 28 )$.\\
(i) Find the values of $\mathrm { E } ( Y - X )$ and $\operatorname { Var } ( Y - X )$.\\
(ii) State, with justification, an approximate distribution for $Y - X$.\\
(iii) Hence find $\mathrm { P } ( | Y - X | \geqslant 3 )$.
\hfill \mbox{\textit{OCR S3 2012 Q5 [10]}}