| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Moment generating function problems |
| Difficulty | Standard +0.8 This is a Further Maths S4 question requiring recognition that the MGF represents a binomial distribution (n=3, p=3/4), differentiation to find E(X), expansion to find P(X=2), and understanding that X is a sum of Bernoulli trials. While systematic, it requires multiple techniques and conceptual understanding of MGFs and their relationship to sums of independent variables, placing it moderately above average difficulty. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(M'(t) = 3(\frac{1}{4} + \frac{3}{4}e^t)^2 \times \frac{3}{4} e^t\) | M1 | Allow one error |
| A1 | ||
| \(E(X) = M'(0) = \frac{9}{4}\) | A1 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| mgf \((\frac{1}{64} + \frac{9}{64}e^t +)\ \frac{27}{64}e^{2t}(+ \frac{27}{64}e^{3t})\) | M1 | Or PGF \(= (\frac{1}{4} + \frac{3}{4}z)^3\) expand, find coefficient of \(z^2\) |
| A1 | ||
| \(P(X=2) =\) coefficient of \(e^{2t} = \frac{27}{64}\) | A1 3 | \(\frac{27}{64}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Sum of 3 obs of \(Y\) with mgf \(\frac{1}{4} + \frac{3}{4}e^t\) has mgf of \(X\) | M1*dep | |
| \(y\): 0, 1 | OR \(B(1, \frac{3}{4})\) | |
| \(p\): \(\frac{1}{4}\), \(\frac{3}{4}\) | A1 | |
| \(\text{Var}(Y) = \frac{3}{4} - (\frac{3}{4})^2 = \frac{3}{16}\) | *M1A1 4 | Using \(E(Y^2) - (E(Y))^2\) OR \(1 \times \frac{3}{4} \times \frac{1}{4}\); M0 if integration used |
| [10] |
## Question 4:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $M'(t) = 3(\frac{1}{4} + \frac{3}{4}e^t)^2 \times \frac{3}{4} e^t$ | M1 | Allow one error |
| | A1 | |
| $E(X) = M'(0) = \frac{9}{4}$ | A1 **3** | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| mgf $(\frac{1}{64} + \frac{9}{64}e^t +)\ \frac{27}{64}e^{2t}(+ \frac{27}{64}e^{3t})$ | M1 | Or PGF $= (\frac{1}{4} + \frac{3}{4}z)^3$ expand, find coefficient of $z^2$ |
| | A1 | |
| $P(X=2) =$ coefficient of $e^{2t} = \frac{27}{64}$ | A1 **3** | $\frac{27}{64}$ |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sum of 3 obs of $Y$ with mgf $\frac{1}{4} + \frac{3}{4}e^t$ has mgf of $X$ | M1*dep | |
| $y$: 0, 1 | | OR $B(1, \frac{3}{4})$ |
| $p$: $\frac{1}{4}$, $\frac{3}{4}$ | A1 | |
| $\text{Var}(Y) = \frac{3}{4} - (\frac{3}{4})^2 = \frac{3}{16}$ | *M1A1 **4** | Using $E(Y^2) - (E(Y))^2$ OR $1 \times \frac{3}{4} \times \frac{1}{4}$; M0 if integration used |
| | **[10]** | |
---4 The discrete random variable $X$ has moment generating function $\left( \frac { 1 } { 4 } + \frac { 3 } { 4 } \mathrm { e } ^ { t } \right) ^ { 3 }$.\\
(i) Find $\mathrm { E } ( X )$.\\
(ii) Find $\mathrm { P } ( X = 2 )$.\\
(iii) Show that $X$ can be expressed as a sum of 3 independent observations of a random variable $Y$. Obtain the probability distribution of $Y$, and the variance of $Y$.
\hfill \mbox{\textit{OCR S4 2011 Q4 [10]}}