1. Multivariate Distributions

2. Distributions The joint distribution of two random variables is f(x 1,x 2 ) The marginal distribution of f(x 1 ) is obtained forgetting (integrating) about the values of x 2 The conditional distribution is obtained fixing the value of one of the variables and looking at the other (x 1 |x 2 ) The variables are independent if f(x 1,x 2 )= f(x 1 ) f(x 2 )

6. Joint Normal and marginals

7. Distributions These ideas generalize for any number of variables X=(x 1,…x p ), f(X)=f(x 1,…x p ), joint f(x 1,…,x r ) si r<p marginal f(x 1,…,x r |x r+1,…,x k ) conditional

8. Curse of dimensionality The space is vide in high dimensions

9. The number of parameters grows faster than the dimension

10. The key variable N/p (data by dimension) At least 10 for inference and if possible 30

13. The normal k dimensional

15. proprieties

16. propiedades

22. Mixtures of distributions

25. Mixture distributions