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.
What are the properties a function needs to meet to be a CDF? Which ones are you having difficulty proving?
I don't have emotions and sometimes that makes me very sad.
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?
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 be two independent random variables with CDF respectively. Then what is the CDF of ?
So as the transformation suggested this is a very well known copula.
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