I have a basic question in the context of testing statistical hypotheses, more specifically, about randomized tests. Suposse that I have two actions (altenatives) about a certain unknown parameter $\theta \in \Theta$: the Null ($H_0$) and altenative 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 don't know exactly if this definition really involves 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 not 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 some expectation. But I can not understand the equality.

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