Why is there an intrinsic trade off between the probability of detection and probability of a false alarm in the operating characteristic? I was reading some notes for fun on Binary Hypothesis testing and it claimed that there happens to be a tradeoff between the probability of detection (also known as the "power"):
$$P_D = P(f(y) = H_1 \mid H = H_1) = \int_{Y_1} p_{y|H}(y \mid H_1) dy$$
and the probability of false alarm (also known as the "size"):
$$P_F = P(f(y) = H_1 \mid H = H_0) = \int_{Y_1} p_{y|H}(y \mid H_0) dy $$ 
and that it was depicted by the Operating Characteristic (OC):

However, it did not explain intuitively nor mathematically rigorously why this trade off occurs nor why the shape of the graph is the way it is. My question is:


*

*Why do $P_D$ and $P_F$ have an intrinsic trade off?

*Why is the shape of the graph the way it is?


For example, why does the graph bend away to the top left corner and away from the line $P_D = -P_F$ and not towards the other direction? i.e. why doesn't the graph bend towards the bottom right corner?
 A: $P_{D}$ is desirable and $P_{F}$ is undesirable. The tradeoff is that you can easily increase $P_{D}$ but at the cost of also increasing $P_{F}$. Consider for example a classifier that always chooses $H_{1}$ regardless of the input. This will always detect positive cases and thus will have a perfect $P_{D}$ of $1$. However, it will also always raise false alarms and thus a perfectly bad $P_{F}$ of $1$. The tradeoff is to maximise $P_{D}$ and simultaneously minimize $P_{F}$. The perfect spot on your chart would be the upper left-hand corner (i.e. $P_{D} = 1$ and $P_{F} = 0$). If your chart bent the other way, that would mean that for every possible $\eta$, the probability of a false alarm is higher than the probability of a true detection so this would be a terribly unreliable classifier. The goal of your OC curve is most likely to choose an optimal $\eta$ which would be the $\eta$ that gets you closest to that ideal upper left corner of your chart.
A: Want to punish all criminals? Throw everyone in jail. Never want to punish an innocent defendant? Don’t ever prosecute anyone.
As you require a higher standard of proof ($\alpha$ or specificity), it’s harder to convict. If you need fingerprints, DNA, and two eyewitness, that’s a lot of evidence compared to just relying on one eyewitness and some prints. You will convict fewer people (low power). 
If you demand higher power, lower your standards for the evidence required to convict (higher $\alpha$). You will convict more innocent defendants, however.
I find sensitivity and specificity to be extremely helpful in describing what $1-\beta$ and $\alpha$ mean, respectively.
$\alpha$ is the type I error rate (specificity). $\beta$ is the type II error rate, so $1-\beta$ is the power (same as sensitivity).
