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I came across this problem as a beginner in probability theory, and I got various answers for it. I hope someone could help me with this.

$F(x)$ and $G(x)$ are CDFs. Is it true that $F(x)G(y)$ and $\frac{F(x)+G(y)}{2}$ are both CDFs too?

I tried to test three things: in minus infinity, it is zero; in infinity it is one; it is monotonically increasing. However, I could not come to terms with this. I found that the product is okay, however, there can be problems with the mean, when trying to integrate with $Y$ as the $F(Y)$ part had disappeared when integrating by $X$.

I would really appreciate any help and confirmal.

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    Just checking, do you intend to say $F(x)G(x)$ or $F(x)G(y)$? – 2017-01-09
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    The latter one! – 2017-01-09
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    Just curious, why do you integrate the CDF? it is to verify certain properties that CDF have? if so, may I know what is this property? – 2017-01-09
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    Well, I raised this question to a friend of mine, and his answer was something like that I was trying to explain, with the integration part, I was hoping it was something important. My initial idea was only to test the (minus) infinity, and the monotonousity properties. – 2017-01-09
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    The function $H:(x,y)\mapsto\frac12(F(x)+G(y))$ is not a joint CDF, for example, $H(x,y)$ does not converge to $0$ when $x\to-\infty$, for every $y$ fixed, as it should. (The other function is trivially a CDF, the simplest way to see that it is might be to exhibit $(X,Y)$ with this joint CDF.) – 2017-01-09

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