# A question about Two-Sample poisson test [closed]

I read this paper: "Experiment Size and Power Comparisons for Two-Sample Poisson Tests", Wei-Kei Shiue and Lee J. Bain, Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 31, No. 2 (1982), pp. 130–134, about which I have a question.

Suppose two Poisson processes are observed for fixed $$s_1$$ and $$s_2$$ respectively,and let $$x$$ and $$y$$ denote the number of outcomes observed.

That is, $$X \sim \operatorname{Poisson}(\lambda_1), \quad Y \sim \operatorname{Poisson}(\lambda_2), \quad\lambda_i=s_ir_i$$Consider a test of $$H_0:r_1=r_2$$ at the $$\alpha$$ significance level against the one-sided alternative $$H_a:r_2>r_1$$.

I understand the conditional distribution of the variable $$Y$$ given the total $$x+y=m$$ is a binomial distribution,$$(Y\mid x+y=m) \sim \operatorname{Bin}(m,p)\,, \quad \text{where} \quad p=\frac{\lambda_2}{\lambda_1+\lambda_2}$$ for the unequal interval case the test of $$H_0:r_1=r_2$$ is equivalent to a binomial test of $$H_0:p=\frac{s_2}{s_1+s_2}=p_0$$.

If $$B(x;n,p)$$ denotes a cumulative binomial distribution with parameters $$n$$ and $$p$$ , then a $$\text{UMPU}$$ test of $$H_0$$ at level $$1-B(c-1;m,p_0)$$ is to reject $$H_0$$ if $$y \geq c$$.

Question: Why may this test also be expressed in terms of Snedecor's $$F$$-distribution as reject $$H_0$$ if $$\frac{(x+1)p_0}{y(1-p_0)}\leq f_\alpha(2y,2(x+1))\,,$$ where $$f_\alpha$$ denotes the $$\alpha$$ percentile of the $$F$$-distribution?

I can't understand what the rejection region is , and why it can expressed in terms of F-distribution.

## closed as off-topic by gung♦, kjetil b halvorsen, Michael Chernick, John, Nick CoxMar 26 '17 at 9:41

This question appears to be off-topic. The users who voted to close gave this specific reason:

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• Please type your question as text, do not just post a photograph (see here). When you retype the question, add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung Mar 25 '17 at 16:56

(I do wish people would stop using $B()$ for the binomial, since that's the natural symbol for the beta - so that's confusing enough; and worse we often -- including right now - want to talk about both, in which case it becomes almost impossible not to be confused when you see capital Beta while you're talking about the beta, but it actually means something else, the binomial.)