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

#1
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.
 
#2
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:

Dason

Ambassador to the humans
#3
What are the properties a function needs to meet to be a CDF? Which ones are you having difficulty proving?
 
#4
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?
 

BGM

TS Contributor
#5
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.