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.