  First we prove some kernel representations for the covariance of two
functions taken on the same random variable and deduce kernel representations
for some functionals of a continuous one-dimensional measure. Then we apply
these formulas to extend Efron's monotonicity property, given in Efron [1965]
and valid for independent log-concave measures, to the case of general measures
on $\mathbb{R}^2$. The new formulas are also used to derive some further
quantitative estimates in Efron's monotonicity property.
