# Modelling probability spaces

1. Apr 14, 2010

### cappadonza

measure spaces and random variable are still fuzzy to me although i'm starting to see how random variable are s
used, i need some clarification on the following

suppose we have a random variable $$X$$ we don't know what the exact sample space is but say we know the density is $$f_{X}(x) , x \in [a,b]$$
now can we re-model the probability space as
$$\Omega = [a,b]$$
sigma field $$\mathcal{F} = \mathcal{B}([a,b])$$ and probability measure $$P(M) = \int_{M}f_{X}(x) dm$$

Can i assume that the borel set generated by [a,b] is a valid sigma field, since $$X$$ has a density defined on [a,b], so the measure must be defined on all $$\mathcal{B}([a,b])$$ as well (not sure how to prove this formally, if this is correct)
suppose now if we had a Random variable $$Y = 2X$$
then we could say the distribution of Y $$P_{Y}(B) = P(\{ Y \in B \}) = \int_{K}f_{X}(x) dm$$ where $$K = \{ k : 2k \in B \}$$.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted