Multivariate Distributions

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)

Joint Normal and marginals

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

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

The normal k dimensional

proprieties

propiedades

Mixtures of distributions

Mixture distributions