Mathematics Prelims

October 26, 2008

Random Variables

Filed under: Measure Theory,Probability Theory,Topology — cjohnson @ 8:32 pm

If (\Omega, \mathcal{F}, P) is a probability space and (Y, \Sigma) is a measurable space (i.e., a set Y along with a sigma-algebra \Sigma on Y), then a random variable is a measurable function X : \Omega \to Y.  That is, for each A \in \Sigma, we have X^{-1}(A) = \{ \omega \in \Omega : X(\omega) \in A \} \in \mathcal{F}.  Generally speaking, we’ll be taking (Y, \Sigma) to be (\mathbb{R}, \mathcal{B}) where \mathcal{B} is the Borel algebra on \mathbb{R}.

Given a topological space (S, \tau), there exists a sigma-algebra \mathcal{B}(S), called the Borel algebra on S, that contains all open sets (members of \tau) and is the smallest such sigma-algebra.  This means that for each \mathcal{O} \in \tau we have \mathcal{O} \in \mathcal{B}(S) and also if \Sigma is another sigma-algebra on S with this property, then \mathcal{B}(S) \subseteq \Sigma.  In general, given a collection C of subsets of S, there exists a sigma-algebra, which we’ll call \mathcal{F}_C, that is the smallest sigma-algebra on S (smallest using \subseteq as the ordering relation) containing each c \in C; we say that \mathcal{F}_C is the sigma-algebra generated by C.  In this sense, the Borel algebra on S is the sigma-algebra generated by the topology \tau.  We will usually just write \mathcal{B} to mean \mathcal{B}(\mathbb{R}).

In the special case of (\mathbb{R}, \mathcal{B}), there’s an easy way to check to see that a function X : \Omega \to \mathbb{R} is a random variable (or a measurable function in general): we just look at the pre-images of intervals.  Since the pre-images of functions are well-behaved with respect to set operations like union, intersection, and complement, it in fact suffices to only consider pre-images of the form {}[a, \infty).  That is, if we show that X^{-1}([a, \infty)) \in \mathcal{F} for every a \in \mathbb{R}, we will have shown that X is measurable, and so a random variable.  (Actually, we can look at all intervals of the form (a, \infty), {}[a, \infty), (-\infty, a) or (-\infty, a].  Using properties of sigma-algebras we can easily show that if we have all intervals of any of these forms, we have all intervals of any other form.  Again, using properties of sigma-algebras it’s easy to take that and show that we have all countable unions of intervals — namely all countable unions of open intervals, i.e., all open sets.)

To see this, suppose for every a \in \mathbb{R} we have X^{-1}([a, \infty)) \in \mathcal{F}.

\displaystyle X^{-1}([a, \infty)) \in \mathcal{F}

\displaystyle \implies \{ \omega \in \Omega : X(\omega) \geq a \} \in \mathcal{F}

\displaystyle \implies \{ \omega \in \Omega : X(\omega) \geq a \}^\complement \in \mathcal{F}

\displaystyle \implies \{ \omega \in \Omega : X(\omega) < a \} \in \mathcal{F}

\displaystyle \implies X^{-1}((-\infty, a)) \in \mathcal{F}

So now we have the pre-images of all intervals of the form {}[a, \infty) and (-\infty, a).  If we can also get pre-images for the form (a, \infty), it’ll be an easy jump to countable unions of open intervals.  Note that

\displaystyle (a, \infty) = \bigcup_{n=1}^\infty \left[ a + \frac{1}{n}, \infty \right)

Now we easily see that

\displaystyle X^{-1}((a, \infty)) = X^{-1} \left( \bigcup_{n=1}^\infty \left[ a + \frac{1}{n}, \infty \right)\right)

\displaystyle = \bigcup_{n=1}^\infty X^{-1}\left(\left[ a + \frac{1}{n}, \infty \right)\right)

And we have pre-images of intervals of the form (a, \infty).  Combined with the fact we have intervals of the form (-\infty, b), it’s easy to see that we also have intervals of the form (a, b).  Using properties of sigma-algebras, it’s easy to show now that we have the pre-image of any open set.  This tells us that if X^{-1}([a, \infty)) \in \mathcal{F} for every a \in \mathbb{R}, then we have that X is measurable, and so a random variable.

In the case that \Omega is a countable set, we take \mathcal{F} to be 2^\Omega (the powerset of \Omega), and so any function X : \Omega \to \mathbb{R} is a random variable.  This is because the pre-image of {}[a, \infty) must be something (even if it’s empty); we have for every a \in \mathbb{R} that X^{-1}([a, \infty)) \in 2^\Omega, and so every function on a countable sample-space is a random variable.

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