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CantorSet
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Hi everyone,
I'm confused about Durrett's formal definition of a random variable, as well his formal notions of probability spaces in general. I always try to make abstract definitions concrete through simple examples, but I can't wrap my head around this one:
Durrett defines:
X is a random variable if X: \Omega \rightarrow \Re and for every Borel set B \subset R, we have X^{-1}(B) \in F, where F is the sigma-algebra on \Omega.
Well, I tried to make this concrete with a simple example. Suppose we have a box with 100 balls labeled from 1 to 100. We draw 3 balls from this box without replacement. Let X be the sum of these three balls. We know that X is a random variable and it takes on values from 3 to 297.
To interpret this example in the context of Durrett's formal definition, I guess \Omega is the set of all subsets of size three from 1 to 100. So \Omega = \{\{1,2,3\} , \{1,2,4\}, ...\} The sigma-algebra F is all collections of these subsets. Then, our random variable X basically only maps the subsets of size 3 to the integers from 3 to 279. So far so good.
But the definition says for every Borel set B \subset R, which we can take as B = (0,1), a perfectly legit Borel set, and we would have X^{-1}(B) \in F. But the problem is the random variable X only maps to discrete points, not intervals. So based on this definition, X wouldn't be a random variable. But I know it is.
Any suggestions on where I may be interpreting things wrong here?
I'm confused about Durrett's formal definition of a random variable, as well his formal notions of probability spaces in general. I always try to make abstract definitions concrete through simple examples, but I can't wrap my head around this one:
Durrett defines:
X is a random variable if X: \Omega \rightarrow \Re and for every Borel set B \subset R, we have X^{-1}(B) \in F, where F is the sigma-algebra on \Omega.
Well, I tried to make this concrete with a simple example. Suppose we have a box with 100 balls labeled from 1 to 100. We draw 3 balls from this box without replacement. Let X be the sum of these three balls. We know that X is a random variable and it takes on values from 3 to 297.
To interpret this example in the context of Durrett's formal definition, I guess \Omega is the set of all subsets of size three from 1 to 100. So \Omega = \{\{1,2,3\} , \{1,2,4\}, ...\} The sigma-algebra F is all collections of these subsets. Then, our random variable X basically only maps the subsets of size 3 to the integers from 3 to 279. So far so good.
But the definition says for every Borel set B \subset R, which we can take as B = (0,1), a perfectly legit Borel set, and we would have X^{-1}(B) \in F. But the problem is the random variable X only maps to discrete points, not intervals. So based on this definition, X wouldn't be a random variable. But I know it is.
Any suggestions on where I may be interpreting things wrong here?
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