Re: Random Variables

On Mar 28, 1:42 pm, Linear1983 <Whitesox2...@xxxxxxx> wrote:
Suppose you want to define a probability space in which the outcomes are not discrete, but any number in the interval [0, 1]. In general terms, what problems do you see, and how would you solve them? (ex: what are events and probabilities?)

In general, a probability space is taken to be a triple (X,B,p), where
X is the set of all outcomes, B is the collection of events, and m is
a function B --> [0,1] that assigns to each event its probability.
There are certain properties that this triple must satisfy: B must be
closed under countable union and complement, m must be countable
additive. If the outcome of a random variable is a real number in
[0,1] chosen "uniformly," then the probability space is usually taken
to be ([0,1], B, m), where B is either the collection of Borel subsets
of [0,1] or the collection of Lebesgue-measurable subsets of [0,1] (I
have seen both), and m is the Lebesgue measure. The thing to watch
out for is that there are subsets of [0,1] which are NOT events. They
are called non-Borel or non-measurable sets (depending on B). The
probability of the random variable taking a value in such a set is not
defined.

I hope this answers your question.

.

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