Find the test that minimizes the sum of the type 1 and type 2 error probabilities

Question

Let we had one observation of $X~ N(\mu,1)$ And consider test : $H_0:\mu=0, H_1:\mu=1$. How to find test that minimizes sum of probablities of type 1 and type 2 errors?

Answer 1

Let's take the definitions of $\alpha$ and $\beta$, type 1 and 2 errors, respectively

$$\alpha=\mathbb{P}[X>z|\mu=0]=1-\Phi(z)$$

$$\beta=\mathbb{P}[X<z|\mu=1]=\Phi(z-1)$$

Now we have to minimize

$$\alpha+\beta=1-\Phi(z)+\Phi(z-1)$$

Let's derive obtaining

$$\frac{d}{d z}(\alpha+\beta)=-\phi(z)+\phi(z-1)=0$$

that implies to solve

$$-e^{-\frac{z^2}{2}}+e^{-\frac{(z-1)^2}{2}}=0$$

which leads immediately to the solution

$$z=\frac{1}{2}$$