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I used the MATLAB code written below to create the following probability density function. It creates the familiar hill-shaped distribution.

bivariate normal distribution

I'm interested to see (whether via MATLAB code or just an linked image) a PDF plot of a distribution that is univariate normal but is multivariate platykurtic or multivariate leptokurtic. Failing that, I would like to read a description of how my image would like different if it was univariate normal but is multivariate platykurtic or multivariate leptokurtic. At present I cannot visualize what that would look like.

MATLAB code:

function prob = csevalnorm(x,mu,cov_mat);
[n,d] = size(x);
% center the data points
x = x-ones(n,1)*mu;
a = (2*pi)^(d/2)*sqrt(det(cov_mat));
arg = diag(x*inv(cov_mat)*x');
prob = exp((-.5)*arg);
prob = prob/a;

% Get the mean and covariance.
mu = zeros(1,2);
cov_mat = eye(2);% Identity matrix
% Get the domain.
% Should range (-4,4) in both directions.
[x,y] = meshgrid(-4:.2:4,-4:.2:4);
% Reshape into the proper format for the function.
X = [x(:),y(:)];
Z = csevalnorm(X,mu,cov_mat);
% Now reshape the matrix for plotting.
z = reshape(Z,size(x));
subplot(1,2,1) % plot the surface
surf(x,y,z),axis square, axis tight
title('BIVARIATE STANDARD NORMAL')
user1205901 - Слава Україні
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    Good question, it is difficult to superimpose 3d (or 2.5d) data in a plot. Perhaps visualize the difference of the densities, or just make two small multiple plots of the different densities with the same perspective. In a 2d graph, I've seen cartographers superimpose contour lines and kde estimates (displayed via color), but they are difficult graphics to interpret for sure. – Andy W May 21 '13 at 13:43
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    Given that kurtosis usually refers to properties of the fourth moment and bivariate distributions have not one but five distinct fourth moments, could you more precisely explain what you mean by "platykurtic" or "leptokurtic" in the bivariate case? – whuber May 21 '13 at 14:26
  • I suspect I'm using the terms imprecisely. I'm seeking a picture of a bivariate distribution that has non-normal kurtosis and don't particularly mind in what way the kurtosis is non-normal. What has occurred is that I've run a test of multivariate kurtosis and a program has told me the results are non-normal. I'm attempting to picture what could potentially look like, or seeking an explanation of why attempting to picture this is impossible in principle. – user1205901 - Слава Україні May 21 '13 at 15:57
  • Perhaps you could give more detail about what it was that led to the question? That might make what you're asking about more clear. – Glen_b May 21 '13 at 23:09
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    See this fantastic answer for a huge collection of bivariate nonnormal distributions that are marginally normal. – Dilip Sarwate May 22 '13 at 02:13
  • @Glen_b Someone asked me to show them a bivariate normal distribution so I showed them that MATLAB plot. Then they asked me to show them a bivariate non-normal distribution so I drew that on a piece of paper. Then they asked me to show them a bivariate non-normal but univariate normal distribution and I had no idea what to draw. – user1205901 - Слава Україні May 22 '13 at 02:58
  • @DilipSarwate Thanks very much for linking that, the heat maps are extremely close to what I was trying to plot. – user1205901 - Слава Україні May 22 '13 at 02:59

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