# Statistical inference – A basic question about a randomized test involving Type I error.

I have a fundamental question in the context of the verification of statistical assumptions, specifically randomized tests. Suposse that I have two actions (alternatives) on a certain unknown parameter $$theta in Theta$$: the null ($$H_0$$) and alternative hypotheses ($$H_1$$).

In this case, the sample space is $$(0,15) subset mathbb {R}$$. We know that the critical function is given by
$$phi (x) = P (reject , , H_0 , , | , , x , , observed)$$

I do not know exactly if this definition really implies a conditional probability. Suposse I have the following critical function

$$phi (x) = begin {cases} 0, quad x in (0,2) \ p, quad x in (2,10) \ 1, quad x in (10,15) \ end {cases}$$

I can understand why

$$P (reject , , H_0 , , , , H_0 , , is , , true) , = 0 times P (x in (0,2)) + p times P (x in (2,10)) + 1 times P (x in (10,15))$$

The right side looks a lot like a wait. But I can not understand.