Add notes for Pd and Pfa

mhostetter · Jun 3, 2024 · 3deae93 · 3deae93
1 parent b2ced48
commit 3deae93
Showing 1 changed file with 12 additions and 0 deletions.
diff --git a/src/sdr/_detection/_theory.py b/src/sdr/_detection/_theory.py
@@ -577,6 +577,12 @@ def p_d(
     Returns:
         The probability of detection $P_d$ in $(0, 1)$.
 
+    Notes:
+        The probability of detection $P_d$ is defined as the probability that the detector output $T(x)$ exceeds the
+        detection threshold $\gamma$ given that the signal is present.
+
+        $$P_d = P\left(T(x) > \gamma \mid \mathcal{H}_1\right)$$
+
     Examples:
         .. ipython:: python
 
@@ -730,6 +736,12 @@ def p_fa(
     Returns:
         The probability of false alarm $P_{fa}$ in $(0, 1)$.
 
+    Notes:
+        The probability of false alarm $P_{fa}$ is defined as the probability that the detector output $T(x)$ exceeds
+        the detection threshold $\gamma$ given that the signal is absent.
+
+        $$P_{fa} = P\left(T(x) > \gamma \mid \mathcal{H}_0\right)$$
+
     Examples:
         .. ipython:: python