MATLAB: Multivariate Nomral Distribution Matlab

joint distributionmultivariatemvnpdfnormal distribution

I want to generate multivariate normal joint distribution for a 3*3 matrix. Say I have mu = [0.4 0.5] and sigma = [0.25 03; 0.45 0.5]. As I wanted to generate a 3*3 joint distribution, I decided to do the following:
mu = [0.4 0.5];
sigma = [0.25 0.3; 0.3 0.5];
x1 = 1:3;
x2 = 1:3;
[X1,X2] = meshgrid(x1,x2);
X = [X1(:) X2(:)];   
y = mvnpdf(X,mu,sigma);
My assumption is that X now has 3*3 matrix index. If the value is 3 1 it means X = 3, Y = 1. For example, if I print the value of X I get:
1     1
1     2
1     3
2     1
2     2
2     3
3     1
3     2
3     3
So, my assumption was the final pdf will be a 9*1 vector and each value will represent the probability of each row. For example, if the second value is 0.05, it will indicate P(X=1,Y=2) = 0.05.
When I gey y, I get the following:
y =
    0.3484
    0.0471
    0.0000
    0.0000
    0.0027
    0.0015
    0.0000
    0.0000
    0.0000
However, as y is a joint distribution, the sum(y) should have been 1. But here it's 0.3997.
Can anyone tell me what am I doing wrong here? And why the sum is not 1?

Best Answer

If you have only those 9 discrete combinations, then you can make their probabilities be any values you want (summing to one). For example, you could use the PDF values and rescale them to sum to one. Whether this is a good idea or not, I cannot say, since I have no idea what you are trying to achieve.
No, there is no representation of Gaussian over the integers 1, 2, and 3.
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