  The bivariate normal density with unit variance and correlation $\rho$ is
well-known. We show that by integrating out $\rho$, the result is a function of
the maximum norm. The Bayesian interpretation of this result is that if we put
a uniform prior over $\rho$, then the marginal bivariate density depends only
on the maximal magnitude of the variables. The square-shaped isodensity contour
of this resulting marginal bivariate density can also be regarded as the
equally-weighted mixture of bivariate normal distributions over all possible
correlation coefficients. This density links to the Khintchine mixture method
of generating random variables. We use this method to construct the higher
dimensional generalizations of this distribution. We further show that for each
dimension, there is a unique multivariate density that is a differentiable
function of the maximum norm and is marginally normal, and the bivariate
density from the integral over $\rho$ is its special case in two dimensions.
