Editing Receiver operating characteristic (section)

===Probabilistic interpretation===
The area under the curve (often referred to as simply the AUC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative').<ref name="fawcett">Fawcett, Tom (2006); ''[https://www.math.ucdavis.edu/~saito/data/roc/fawcett-roc.pdf An introduction to ROC analysis]'', Pattern Recognition Letters, 27, 861–874.</ref> In other words, when given one randomly selected positive instance and one randomly selected negative instance, AUC is the probability that the classifier will be able to tell which one is which.

This can be seen as follows: the area under the curve is given by (the integral boundaries are reversed as large threshold <math> T </math> has a lower value on the ''x''-axis)
:<math>\operatorname{TPR}(T): T \to y(x)</math>
:<math>\operatorname{FPR}(T): T \to x</math>
:<math>
\begin{align}
A & = \int_{x=0}^1 \mbox{TPR}(\mbox{FPR}^{-1}(x)) \, dx \\[5pt]
& = \int_{\infty}^{-\infty} \mbox{TPR}(T) \mbox{FPR}'(T) \, dT \\[5pt]
& = \int_{-\infty}^\infty \int_{-\infty}^\infty I(T' \ge T)f_1(T') f_0(T) \, dT' \, dT = P(X_1 \ge X_0)
\end{align}
</math>
where <math> X_1 </math> is the score for a positive instance and <math> X_0 </math> is the score for a negative instance, and <math>f_0</math> and <math>f_1</math> are probability densities as defined in previous section.

If <math> X_0 </math> and <math> X_1 </math> follows two Gaussian distributions, then <math> A = \Phi\left((\mu_1-\mu_0)/\sqrt{\sigma_1^2 + \sigma_0^2}\right) </math>.