In chapter 5, at page 47, when speaking about the properties of maximum likelihood estimators of a time-homogenous Markov jump process, it says: "the components are uncorrelated and so independent (being Normal)". I understand this as: if two random variable are normally distributed and are uncorrelated, they must be independent. Can somebody explain me this (why two normal and uncorrelated components are independent)?
I believe this is only true if the variables are jointly normal. Given this, if they are uncorrelated the value of the correlation coefficient (rho) of the joint pdf becomes zero. If rho is zero the joint normal pdf simplifies from: f(x,y) = f(x) . f(y) + correlation effect to f(x,y) = f(x) . f(y) implying independence.
