Multivariate Distributions. Distributions The joint distribution of two random variables is f(x 1,x 2 ) The marginal distribution of f(x 1 ) is obtained.

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Presentation transcript:

Multivariate Distributions

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 )

Joint Normal and marginals

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

Curse of dimensionality The space is vide in high dimensions

The number of parameters grows faster than the dimension

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

The normal k dimensional

proprieties

propiedades

Mixtures of distributions

Mixture distributions