What must be true before adding Poisson means?

The variables must be independent.

No. Only sums of independent Poisson variables are Poisson.

What is H0 in a Poisson test?

H0:\lambda=\lambda0, a stated value of the mean.

What data does the test in this topic use?

A single observation of the count.

One-tailed test, H1:\lambda\lambda0, observation x. What do you calculate?

P(X\ge x)=1-P(X\le x-1) under H0.

When do you reject H0?

When the probability is less than or equal to the significance level.

What does 'significance level' mean?

The probability threshold below which the result is judged too unlikely under H0.

Wording of a conclusion when H0 is not rejected?

'Insufficient evidence at the 5% level that...' (in context).

Why is \lambda defined in the hypotheses?

So the hypotheses are clearly about the population mean in context.

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

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Distribution of $X+Y$ for independent $X\sim\mathrm{Po}(\lambda)$, $Y\sim\mathrm{Po}(\mu)$?

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Distribution of $X+Y$ for independent $X\sim\mathrm{Po}(\lambda)$ , $Y\sim\mathrm{Po}(\mu)$ ?: $\mathrm{Po}(\lambda+\mu)$
What must be true before adding Poisson means?: The variables must be independent.
Is $X-Y$ Poisson?: No. Only sums of independent Poisson variables are Poisson.
Total over 4 weeks if weekly count is $\mathrm{Po}(4)$ ?: $\mathrm{Po}(16)$
What is $\mathrm{H}_0$ in a Poisson test?: $\mathrm{H}_0:\lambda=\lambda_0$ , a stated value of the mean.
What data does the test in this topic use?: A single observation of the count.
One-tailed test, $\mathrm{H}_1:\lambda<\lambda_0$ , observation $x$ . What do you calculate?: $\mathrm{P}(X\le x)$ under $\mathrm{H}_0$ .
One-tailed test, $\mathrm{H}_1:\lambda>\lambda_0$ , observation $x$ . What do you calculate?: $\mathrm{P}(X\ge x)=1-\mathrm{P}(X\le x-1)$ under $\mathrm{H}_0$ .
When do you reject $\mathrm{H}_0$ ?: When the probability is less than or equal to the significance level.
Two-tailed test at 5%: what do you compare with?: $0.025$ in the relevant tail.
What does 'significance level' mean?: The probability threshold below which the result is judged too unlikely under $\mathrm{H}_0$ .
Wording of a conclusion when $\mathrm{H}_0$ is not rejected?: 'Insufficient evidence at the 5% level that...' (in context).
Why is $\lambda$ defined in the hypotheses?: So the hypotheses are clearly about the population mean in context.

Exam questions on Sums of Poisson variables and Poisson hypothesis tests

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
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
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).