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Thread: Proof the sum of two CDF's is a CDF.

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    Proof the sum of two CDF's is a CDF.




    This is a question my instructor asked in the last midterm exam but nobody was able to solve and he's suggesting it may come out again in the finals:

    If F(x) and G(X) are two CDF's, prove that H(X) is also a CDF if
    H(X) = F(X)+G(X)-F(x)G(X)

    He said something about right continuity but I do not know how that helps.

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    Re: Proof the sum of two CDF's is a CDF.

    This looks like the inclusion-exclusion principle for probability if those RV's are independent.
    Recall \displaystyle P\left( A\cup B \right) = P(A)+P(B)-P\left( A \cap B \right)= P(A) + P(B) - P(A)P(B) if they are independent.
    Last edited by JohnK; 10-18-2013 at 05:41 PM.

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    Re: Proof the sum of two CDF's is a CDF.

    What are the properties a function needs to meet to be a CDF? Which ones are you having difficulty proving?
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    Re: Proof the sum of two CDF's is a CDF.

    Properties are:
    1. F is non-decreasing
    2. lim x approaches infinity F(x) = 1
    3. lim x approaches negative infinity F(x) = 0
    4. F is right-continuous

    How can these be applied here?

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    Re: Proof the sum of two CDF's is a CDF.


    If I have not remember wrongly the properties you listed should be the necessary and sufficient condition for a function to be a CDF. So you may check that.

    A smarter way, as JohnK suggested, is the following:

    Let X, Y be two independent random variables with CDF F, G respectively. Then what is the CDF of \min\{X, Y\}?

    So as the transformation suggested this is a very well known copula.

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