Sorry if the question is ill-phrased, but I'm pretty new to this and I have the following situation:
I know that samples X is drawn from a Gaussian Distribution G(u, v). Now that if u is substituted with another Gaussian random variable, what would this conjugate distribution be? Would it still be a Gaussian?
Just to make this more clear, given the following experiment:
- sample a set of points Xs from a normal distribution with known mean and var,
- generate multiple normal distributions Ds where each D has one of Xs being mean and all Ds share the same known var,
- sample a set of Ys from each D, and combine the Ys together.
I'm basically trying to get a clue on what the distribution of Ys is now, is it still Gaussian, what would the mean and var be?
Aaron's demo in his answer is pretty clear. Although in my study, I do tend to use pretty large n and m, per my own experiments, the shape of binning on the resulting points is still pretty Gaussian-like, it got me wondering again if there's theoretical proof behind that given large n and m, it is guaranteed to be Gaussian?
The above graph is generated with n=1000, m=10000, mean=0.95, std_dev1=0.07, std_dev2=0.7