- Given a random vector, what are the
domain, range and sigma algebras on
them for each of its components to
be a random variable i.e. measurable
- is the domain of each component random variable same as the domain of the random vector, and are the sigma algebras on their domains also the same?
- Is the range of each component random variable the component
space in the cartesian product
for the range of the random
vector? What is the sigma algebra
on the range of each component
random variable and how is it
induced from the sigma algebra on
the range of the random vector?
- Please correct me if I am wrong. If I understand correctly, given a random vector, the probability measure induced on the range (which is a Cartesian product space) by the random vector, is called the joint probability measure of the random vector. The probability measure induced on the component space of the range by each component of the random vector, is called the marginal probability measure of the component random variable of the random vector.
- Consider the concept of the component random variables of a random vector being independent. I read from a webpage that it is said so when the joint probability measure is the product measure of the individual marginal probability measures. I was wondering if the sigma algebra for the joint probability must be the same as the product sigma algebra for the individual probability measures, or the former can just contain the latter?
Thanks and regards!