Question

I am reading Elements of Statistical Learning ESLII and in chapter 2, they have a gaussian mixture data set to illustrate some learning algorithms. To generate this data set, they first generate 10 means from a bivariate gaussian distribution N((1,0)’, I). I am not sure what they mean?

How can you generate 10 means from a bivariate distribution having mean(1,0)?

Answer 1

Each of the means that are generated from the bivariate Gaussian distribution are simply single points sampled in exactly the same way as any other random points that could be generated from the distribution. The fact that they use these generated points to be the means of new distributions is irrelevant.

Let’s say that each of the 10 means is then used to construct a new bivariate Gaussian.

means ~ N( (1,0), I)

Where ~ indicates a value being drawn from the distribution. Since the distribution being sampled from in this case is a bivariate Gaussian, each of the data points sampled will be a 2-dimensional point (x1, y1).

Each of these points sampled from the original distribution can then be used to make a new distribution.

Example:

means = [ (x1,y1), (x2,y2), ..., (x10,y10) ]

To build new bivariate Gaussians:

N1((x1,x2), I), N2((x2,y2), I), ..., N10((x10,y10), I)

They are just using the initial bivariate Gaussian distribution N((1,0), I) as an easy way to pick 10 random means that are distributed normally.