Exercise $3\mathrm{.}21.$ Show that precision can be expressed as

$=\frac{{}_1}{{}_1+{}_0}=\frac{{}_1}{(-){}_1+}.$

Given a family of classifiers parametrized by threshold $$ as in $(3\mathrm{.}20),$ we obtain

a parametric collection of precisions, one for each $,$

$()=P(Y=1|hat(Y{)}_{}=1).$

The PR curve is the plot of the parameteric curve $((),())$ where $()$ is

the parametric TPR given in $(3\mathrm{.}22).$ Note that precision is plotted on the $y-$axis

versus TPR on the $x-$axis.

$()$:$=P(hat(Y{)}_{}=1|Y=1),()$:$=P(hat(Y{)}_{}=1|Y=0).$