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Multivariate Distributions

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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Page 1: 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

Multivariate Distributions

Page 2: 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

Distributions

• The joint distribution of two random variables is f(x1,x2)

• The marginal distribution of f(x1) is obtained forgetting (integrating) about the values of x2

• The conditional distribution is obtained fixing the value of one of the variables and looking at the other (x1 |x2)

• The variables are independent if

f(x1,x2)= f(x1) f(x2)

Page 3: 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
Page 4: 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
Page 5: 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
Page 6: 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

Joint Normal and marginals

Page 7: 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

Distributions

• These ideas generalize for any number of variables X=(x1,…xp),

• f(X)=f(x1,…xp), joint

• f(x1,…,xr) si r<p marginal

• f(x1,…,xr |xr+1,…,xk) conditional

Page 8: 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

Curse of dimensionality

• The space is vide in high dimensions

Page 9: 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

The number of parameters grows faster than the dimension

Page 10: 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

• The key variable

N/p (data by dimension)

• At least 10 for inference and if possible 30

Page 11: 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
Page 12: 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
Page 13: 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

The normal k dimensional

Page 14: 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

The normal k dimensional

Page 15: 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

proprieties

Page 16: 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

propiedades

Page 17: 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
Page 18: 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
Page 19: 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
Page 20: 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
Page 21: 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
Page 22: 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

Mixtures of distributions

Page 23: 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
Page 24: 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
Page 25: 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

Mixture distributions

Page 26: 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
Page 27: 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