Question
I want to learn Multivariate Gussian distribution so I written the following code.*I am implementing following formula Iam expecting this type of plot Where I am doing mistake? clear all clc % Taking two guassian random variables x=randn(1000,1); y=randn(1000,1); X=[x y]; X=X'; d=size(X,1); % find means of x,y mx=mean(x); my=mean(y); mumat=[mx my]'; mumat=repmat(mumat,1,size(X,2)); Dif_mat=X-mumat; % The above step (Dif_mat) is (X-mu) in the formula cov_mat=cov(X'); % covariance matrix det_cv=det(cov_mat); % det of cov matrix inv_cov=inv(cov_mat); % inverse of cov matrix % scale term before exp in forumala scale=((2*pi)^(d/2))*sqrt((abs(det_cv))); scale=inv(scale); % Mahabolis distance in formula MB=Dif_mat'*cov_mat*Dif_mat; % find the final probability p=scale*exp((-1/2)*MB); surf(x,y,p)
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Expert Answer
Neeta Dsouza
PhD Expert
Answered Aug 18, 2026
You're really trying to do two things here. The first is, you have some random data and you want to fit it to a multivariate normal distribution. Your approach to this part works, although it can be streamlined:
n = 1000; d=2; X = randn(n,2);
Get mean and covariance:
mumat=mean(X); cov_mat=cov(X);
The second part is plotting the resulting distribution. Here you need a regular grid for your variables, not the random values you generated above:
x = -3:.2:3; y = -3:.2:3; [X,Y] = meshgrid(x,y); X = X-mumat(1); Y = Y-mumat(2);
Combine X and Y in a way that each row represents one 2D variable.
Z = [X(:) Y(:)];
Now calculate the probabilities.
scale=((2*pi)^(d/2))*sqrt(abs(det(cov_mat))); p = zeros(length(Z),1); for ii=1:length(Z) p(ii) = exp(-Z(ii,:)/cov_mat*Z(ii,:)'/2)/scale; end
Reshape and plot.
p = reshape(p,length(x),length(y));
surf(x,y,p)
xlabel('x'), ylabel('y')
For example
Lets Compute and plot the pdf of a bivariate normal distribution with parameters mu = [0 0] and Sigma = [0.25 0.3; 0.3 1].
Define the parameters mu and Sigma.
mu = [0 0]; Sigma = [0.25 0.3; 0.3 1];%initialisation
Now Creating a grid of evenly spaced points in two-dimensional space
x1 = -3:0.2:3; x2 = -3:0.2:3; [X1,X2] = meshgrid(x1,x2); X = [X1(:) X2(:)];
Now we will evaluate the pdf of the normal distribution at the grid points.
y = mvnpdf(X,mu,Sigma); y = reshape(y,length(x2),length(x1));
Now plotting
surf(x1,x2,y)
caxis([min(y(:))-0.5*range(y(:)),max(y(:))])
axis([-3 3 -3 3 0 0.4])
xlabel('x1')
ylabel('x2')
zlabel('Probability Density')
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