# ROC analysis: detection probability when false alarm is maximized.

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...?

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