Recently on Marilyn's discussion boards...
From Gofer:>Why is it that Gofer keeps insisting that a sample sample is something that must be "constructed"? And that I insist it "MUST be constructed from random variables, but that the sample space is constructed first instead and only then a random variable is defined on it, just like Wikipedia states and has multiple examples of?
Is Jeff really suggesting that the problem be solved without actually "constructing" the solution mathematically?
> A sample space is an abstract set that includes all possible outcomes.
No, it really isn't! Because mathematical functions could later be defined on it, just like Wikipedia says!
> We can only ***DESCRIBE*** these outcomes by mapping them in a set of measures. These maps are random variables. specifying them does not ***CONSTRUCT*** the space, which exists one its own, it only constructs the ***DESCRIPTION.***
This is gibberish!
> And the reason poor Gofer can't find this on the internet, is because he is looking in the formal definitions that apply to sample space in general, not to specific examples.
Just too bad that "specific examples" was exactly what Gofer was looking at, more specifically those related to coin tosses, on the Wikipedia article on "random variables".
> So once again, I challenge poor Gopher to describe the sample space for a coin flip without using labels and measures that can be considered - whether or not he calls them such - random variables. If he tries this simple exercise with honest intent, he will see what I mean.
More gibberish! The mapping comes from stating that "Tails" represents landing on tails, and "Heads" landing on heads. Those are the values we put in a set, and call it our sample space, in the language of mathematics.
> A mapping can be a mathematical function being an arithmatical one.
Yes it could! But it needn't be!
>> A mathematical function only operates on well-defined sets.
> The set of all possible outcomes" is well defined, just not described until you describe it with random variables.
I probably should have written "well-constructed" instead of "well-defined".
No, "the set of all possible outcomes" is NOT well defined, but is a description of what you mean, just like:
"Let (f g x) be a function that outputs true or false whether the finite-state machine g halts for input x."
is a description, but is not well constructed, or well defined if you will.
PS one should always strive to be a constructivist, but loosen up here and there DS