A recent question on X validated was looking rather naïvely for the median being equal to the mean, as earlier questions there, but this made me realise a certain symmetry in the two notions, namely that the median ς is solution of the balance condition
namely the areas under the density are equal, while the mean μ is solution of a parallel balance condition
which is an equality of the same kind, i.e., of areas under the cdf or its complement…. Courtesy of the general representation
![\mathbb E[X] = \int_0^{\infty} (1-F)(x) \text dx - \int_{-\infty}^0 F(x) \text dx](https://s0.wp.com/latex.php?latex=%5Cmathbb+E%5BX%5D+%3D+%5Cint_0%5E%7B%5Cinfty%7D+%281-F%29%28x%29+%5Ctext+dx+-+%5Cint_%7B-%5Cinfty%7D%5E0+F%28x%29+%5Ctext+dx&bg=000000&fg=B0B0B0&s=0&c=20201002)
The general problem of having the mean and median equal is not particularly interesting as it is a matter of parameterisation. For instance, using the cdf transform turns the random variate X into a Uniform F(X), with mean and median equal to ½.
