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