Receiver Operating Characteristics

$$P_D(P_{FA})=Q\left(Q^{-1}(P_{FA})-\frac{\mu }{\sigma }\right)$$

Properties

• Always concave downwards
  • This is due to a randomized test is always a straight line. Concave upwards is therefore useless.
• Above the line $P_D=P_{FA}$
• The slope in a given point equals the threshold to achieve power and size of that point $$ \frac{\partial P_{D}}{\partial P_{FA}} = \gamma

## Intuitive Understanding ![[Pasted image 20240221123433.png]] We want a distribution like the yellow one. It is a better detector. This is due to the hypothesizes are getting shifted away from each other, *making it easier to distinguish.*