Confusion Matrix for Hypothesis Testing

Binary Hypothesis case

We look at the following case Example 1 - Change in Mean Detection

$P_{FA}$ is the probability of false alarm

$$\begin{array}{rl}{P}_{FA}& =Pr\{x>\gamma |H_0\}\\ & =Pr\{\mathcal{N}(\mu ,{\sigma }^2)>\gamma \}\\ & =Q\left(\frac{\gamma }{\sigma }\right)\end{array}$$

$P_D$ is the probability of the correct detection

$$P_D=Pr\{x<\gamma |H_1\}=Q\left(\frac{\gamma -\mu }{\sigma }\right)$$