Assume I already have n samples of a 2D variable. I can compute the sample mean and variance. If I assume that the samples are taken from a normal distribution, then using the mean and variance I get a Gaussian g, which is an estimation of that distribution.
Now, I want to calculate the expected estimated distribution, if I take another sample. I assume that the sample is taken from g. It seems like the mean would not change. What happens to the covariance matrix? Is computing the expected new estimated distribution the same as adding a sample whose value is the mean?