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    Probability definition




    Hi stats guys,

    I had a heated discussion about the following situation today:

    Letís say I have a function f(x) producing values between 0 and 1, then I can consider this as a probability. If I have a function g(x) producing values between 0 and 0.5, is this still a probability, since values large than 0.5 could never be achieved?

    Thanks!

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    Re: Probability definition

    Your question doesn't really make sense as is.
    I don't have emotions and sometimes that makes me very sad.

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    Re: Probability definition

    Quote Originally Posted by Dason View Post
    Your question doesn't really make sense as is.
    Okay I try to rephrase:

    1. Can I consider any function f(x) with codomain [0,1] as a probability?

    2. If yes, is a function g(x)=0.5*f(x) with codomain [0,0.5] also a probability?

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    Re: Probability definition

    Well if you really want to interpret the output as a probability then you can do that. Still doesn't necessarily make sense and we wouldn't call it a probability function unless it met other criteria as well.
    I don't have emotions and sometimes that makes me very sad.

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    Re: Probability definition

    Okay thanks for your answer but could you please be more detailed. Sorry that I repeat my question again:

    I am not talking about a PDF (with unity of the integral etc), simply a mechanism (whatever it is) producing values from 0 to 1. To my understanding this output could be considered as a probability, right? If I now modify this function to produce only values from 0 to 0.5, is this is still a probability even values from (0.5,1] never occur.

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    Re: Probability definition

    It would be just about as much as a "probability" as the outputs from the other function.
    I don't have emotions and sometimes that makes me very sad.

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    Re: Probability definition


    To check whether something is a valid probability, you look at the probability axioms and see if they are really satisfied.

    http://en.wikipedia.org/wiki/Probability_axioms

    One thing to note that is P(\Omega) = 1. If your arbitrary function never map to the value 1, then it cannot be a probability.

    Put it in other words: In the probability axioms, there is no requirement that the probability must be a surjective function - as you seen from those simple discrete probability models. However it does require it to map to the values \{0, 1\} as \{\varnothing, \Omega\} is the trival \sigma-algebra over any sample space.

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