| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Multiple independent coins/dice |
| Difficulty | Standard +0.3 This is a straightforward application of standard PGF techniques for Further Maths S4. Part (i) requires constructing a PGF for two independent biased coins (routine), (ii) uses the independence property G_Z = G_X × G_Y (standard result), (iii) applies standard formulas E(Z) = G'(1) and Var(Z) = G''(1) + G'(1) - [G'(1)]², and (iv) simply reads off a coefficient. All steps are mechanical applications of learned techniques with no novel problem-solving required. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^25.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(0) = 0.16\), \(P(1) = 0.48\), \(P(2) = 0.36\) | B1 | |
| \(G_X(t) = 0.16 + 0.48t + 0.36t^2\) | M1A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G_X(t) \times G_Y(t)\) soi | M1 | |
| \(0.02 + 0.12t + 0.285t^2 + 0.335t^3 + 0.195t^4 + 0.045t^5\) | A1A1 | At least 4 terms correct; All correct |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(Z) = G_Z'(1)\ [= 0.12 + 0.57t + 1.005t^2 + 0.78t^3 + 0.225t^4]\) | M1 | Differentiate |
| Sub \(t=1\) | M1dep | |
| \(2.7\) | A1 | |
| Attempt 2nd derivative of \(G_Z\) | M1 | |
| Attempt \(G''(1) + G'(1) - (G'(1))^2\ (G''(1) = 5.82)\) | M1dep | -ve var, M0 |
| \(1.23\) | A1 | |
| Alternative methods: \(3\times0.5\) or \(2\times0.6\); added: \(2.7\) M1M1A1; \(3\times0.5\times0.5\) or \(2\times0.6\times0.4\); added: \(1.23\) M1M1A1 | ||
| \(P(Z=0)=0.02\) etc B1; \(E(Z)=\Sigma zp=2.7\) M1A1; \(E(Z^2)=\Sigma z^2p=(8.52)\) M1; \(-2.7^2\) M1; \(1.23\) A1 | ||
| [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.335\) | B1ft | Coeff \(t^3\) from (ii) |
| [1] |
# Question 6:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(0) = 0.16$, $P(1) = 0.48$, $P(2) = 0.36$ | B1 | |
| $G_X(t) = 0.16 + 0.48t + 0.36t^2$ | M1A1 | |
| **[3]** | | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G_X(t) \times G_Y(t)$ soi | M1 | |
| $0.02 + 0.12t + 0.285t^2 + 0.335t^3 + 0.195t^4 + 0.045t^5$ | A1A1 | At least 4 terms correct; All correct |
| **[3]** | | |
## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(Z) = G_Z'(1)\ [= 0.12 + 0.57t + 1.005t^2 + 0.78t^3 + 0.225t^4]$ | M1 | Differentiate |
| Sub $t=1$ | M1dep | |
| $2.7$ | A1 | |
| Attempt 2nd derivative of $G_Z$ | M1 | |
| Attempt $G''(1) + G'(1) - (G'(1))^2\ (G''(1) = 5.82)$ | M1dep | -ve var, M0 |
| $1.23$ | A1 | |
| Alternative methods: $3\times0.5$ or $2\times0.6$; added: $2.7$ M1M1A1; $3\times0.5\times0.5$ or $2\times0.6\times0.4$; added: $1.23$ M1M1A1 | | |
| $P(Z=0)=0.02$ etc B1; $E(Z)=\Sigma zp=2.7$ M1A1; $E(Z^2)=\Sigma z^2p=(8.52)$ M1; $-2.7^2$ M1; $1.23$ A1 | | |
| **[6]** | | |
## Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.335$ | B1ft | Coeff $t^3$ from (ii) |
| **[1]** | | |
---6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is $\frac { 3 } { 5 }$.
Andrew tosses all five coins. It is given that the probability generating function of $X$, the number of heads obtained on the unbiased coins, is $\mathrm { G } _ { X } ( t )$, where
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$
(i) Find $G _ { Y } ( \mathrm { t } )$, the probability generating function of $Y$, the number of heads on the biased coins.\\
(ii) The random variable $Z$ is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of $Z$, giving your answer as a polynomial.\\
(iii) Find $\mathrm { E } ( Z )$ and $\operatorname { Var } ( Z )$.\\
(iv) Write down the value of $\mathrm { P } ( Z = 3 )$.
\hfill \mbox{\textit{OCR S4 2016 Q6 [13]}}