Sums of Poisson variables and Poisson hypothesis testsAQA A-Level Further Maths: Mind map

Exam questions on Sums of Poisson variables and Poisson hypothesis tests

1. Emails arrive at two servers independently of one another. The number of emails arriving at server A in a minute is $X\sim\mathrm{Po}(3)$ and the number arriving at server B in a minute is $Y\sim\mathrm{Po}(2)$.
   Find the probability that at least 2 emails arrive at the two servers together in a minute.2 marks
2. The number of complaints received by a café in a week has historically followed $\mathrm{Po}(4)$. After staff training, the manager claims that the mean number of complaints per week, $\lambda$, has fallen. In the first week after the training the café receives 1 complaint. A hypothesis test is carried out at the 5% significance level.
   State the conclusion of the test, in context, using your answer to (b).2 marks
3. In a textile factory, flaws in fabric occur at random. Machine A produces flaws at a mean rate of 2.1 per metre and machine B at a mean rate of 1.4 per metre. The numbers of flaws per metre are modelled by $X\sim\mathrm{Po}(2.1)$ for machine A and $Y\sim\mathrm{Po}(1.4)$ for machine B, with $X$ and $Y$ independent.
   One metre of fabric is taken from each machine. Find the probability that the two pieces together have exactly 2 flaws.3 marks