| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Topic | Poisson distribution |
| Type | Conditional probability with Poisson |
| Difficulty | Challenging +1.3 Part (i) is a standard PGF derivation requiring manipulation of the exponential series. Part (ii) is a textbook application of PGF properties for sums of independent variables. Part (iii) requires conditional probability with Poisson distributions, which is more sophisticated but follows a standard technique (conditioning on the sum gives a binomial distribution). This is moderately challenging for Further Maths students but uses well-established methods without requiring novel insight. |
| Spec | 2.03d Calculate conditional probability: from first principles5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02n Sum of Poisson variables: is Poisson |
**Question 4(i)**
$P(X = r) = e^{-\lambda} \frac{\lambda^r}{r!}$ (may be implied by next line.) **B1**
$G_X(t) = \sum_0^{\infty} p_r t^r = \sum_0^{\infty} e^{-\lambda} \frac{(\lambda t)^r}{r!} = e^{-\lambda} e^{\lambda t} = e^{\lambda(t-1)}$ (AG) **M1A1** [3]
**Question 4(ii)**
$G_{X+Y}(t) = e^{\lambda(t-1)} \cdot e^{\mu(t-1)} = e^{(\lambda+\mu)(t-1)} \Rightarrow (X+Y) \sim \text{Po}(\lambda + \mu)$ **M1A1** [2]
**Question 4(iii)**
$P(\,[0,4]\text{ or }[1,3]\text{ or }[2,2]\,) = 0.2231 \times 0.1336 + 0.3347 \times 0.2138 + 0.2510 \times 0.2565$ **B2,1,0**
$P(X \leq 2 \mid X + Y = 4) = \frac{0.1657}{0.1954}$ **M1A1↓**
$= 0.848$ (AWRT) **A1** [5]
(S.C. If no marks earned in **(iii)**, award B1 for $P(X \leq 2) = 0.1657$ if seen.) **[10]**4 (i) The random variable $X$ has the distribution $\operatorname { Po } ( \lambda )$. Prove that the probability generating function, $\mathrm { G } _ { X } ( t )$, is given by
$$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
(ii) The independent random variables $X$ and $Y$ have distributions $\operatorname { Po } ( \lambda )$ and $\operatorname { Po } ( \mu )$ respectively. Use probability generating functions to show that the distribution of $X + Y$ is $\operatorname { Po } ( \lambda + \mu )$.\\
(iii) Given that $X \sim \operatorname { Po } ( 1.5 )$ and $Y \sim \operatorname { Po } ( 2.5 )$, find $\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2012 Q4 [10]}}