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Math Help - Probability Measure Proof

  1. #1
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    Probability Measure Proof

    Q: Let P be some probability measure on sample space S = [0,1].
    a) Prove that we must have lim n->infinity P((0, 1/n)) = 0
    b) Show by example that we might have lim n-> infinity P([0, 1/n)) > 0

    Approach:
    I'm having a really hard time with this one. By looking at the question, it becomes clear that the open/closed interval on 0 must be key to the solution, but I cannot figure why.

    For a), I was going to try and show that lim n->infinity of 1/n -> 0, however, since there is an open interval around 0, it is not actually included and therefore the probability cannot be greater than 0. Is there some mathematical way of portraying this?

    For b), I was thinking of the basic example of flipping a coin and looking at the probability of getting at least 1 head in n tries. I would define the probability function as such:

    P({tails}) = 1 if there are no heads
    = 1/n if there are n heads

    Does this make sense? Any help would be greatly appreciated as I'm almost done my assignment.
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  2. #2
    Moo
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    Hello,

    Sorry to reply so late.
    Consider A=\bigcap_{n\geq 1} (0,\frac 1n). It is possible to prove that there is no element belonging to this intersection : Let x>0 any point in A. By the archimedean property of \mathbb R (I guess it's the word), there exists N such that x>1/N>0. So x\not\in (0,\frac 1N) hence x\not\in A. So A is empty (0 doesn't belong to the set) and P(A)=0.

    And since (0,\frac 1n)\supset (0,\frac{1}{n+1}), A is the intersection of a decreasing sequence of set. So by a property of a measure, 0=P(A)=\lim_{n\to\infty} P((0,1/n))
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