A note on the expected probability of constraint violation in sampled convex programs

In this note we derive an exact expression for the expected probability V of constraint violation in a sampled convex program (see [1], [2] for definitions and an introduction to this topic): This result (Theorem 1) is obtained using a simple technique based on cardinality count. In the note, we also use a Chernoff bounding technique on the upper tail violation probability expression derived in [4] to obtain one of the tightest available explicit bounds on the sample complexity of sampled convex programs (Proposition 3).