Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am wondering if detection probability always goes to 1 as false alarm probability goes to 1.

Let's assume binary hypothesis problem:

$\mathcal{H}_0: x(t) =n(t)$

$\mathcal{H}_1: x(t) = s(t) + n(t)$ where $s(t)$ and $n(t)$ are the desired signal and arbitrary noise respectively.

And we have the two probability density: $p(x|\mathcal{H}_0)$ and $p(x|\mathcal{H}_1)$ with a threshold $\gamma$ where the threshold constrains the false alarm probability.

If the two densities are identical, the ROC curve shows strain line with slop=1.

If $p(x|\mathcal{H}_1)$ has larger variance than $p(x|\mathcal{H}_0)$ and has smaller mean than $p(x|\mathcal{H}_0)$, I thought the detection probability could be less than 1 even though the false alarm probability is 1.

Am I wrong...?

share|cite|improve this question

You cannot have a false alarm (false positive) without a detection (positive). So the answer to your initial query (detection probability to one as FA probability to one) is yes.

(edited because the only "question" you wrote was "Am I wrong..." and I didn't want to seem callous).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.