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Given $c \in R$, a deterministic probability density $f(x)$ and its cumulative distribution $F(c)$, what can be said about $G(c)$ where:

$G(c)=\int f(x)F\left( x+c\right) dx $

The question specifically is:

A) Whether $G(x)$ is concave or convex in $c$?

B) Whether $G(0)=0$ (i.e. $G(c=0)=0.5$)?

C) The sign of the first and second order derivatives wrt $c$.

Any suggestion will be highly appreciated.

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    "Cumulative density" is an oxymoron: the word "cumulative" contradicts the word "density". I presume you mean the cumulative distribution function. That's what you ought to write.2012-10-12
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    you are right, I got it wrong2012-10-15

1 Answers 1

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There is not enough information to answer all of your questions; it depends on the behaviour of $f$. Perhaps useful is the observation that:

$$F(x) = \int_{-\infty}^x f(y)\ \mathrm dy, G(c) = \int_{\Bbb R}\int_{-\infty}^{x+c} f(x)f(y)\ \mathrm dy \mathrm dx$$

Assuming a sufficiently smooth density $f$ is used, we also have:

$$G'(c) = \int_{\Bbb R} f(x)f(x+c) \ \mathrm dx$$

Since $f(x) \ge 0$, $G'(c) \ge 0$ for all $c$. Note that it is not even a priori clear that this integral will exist.

Upon differentiating this expression again we get a term $f'(x+c)$ which can behave in arbitrary ways. Noteworthy is that since $0 \le F(x) \le 1$ for all $x$, we must have $0 \le G(c) \le 1$ for all $c \in \Bbb R$.

Now let us look at the domain of the double integration for $G(c)$. We have that it is:

$$\{(x,y) \in \Bbb R^2: y \le x + c\} = \{(x,y) \in \Bbb R^2: x \ge y - c\}$$

Thus we can rewrite the integral to:

$$\int_{\Bbb R} \int_{y - c}^\infty f(x)f(y) \ \mathrm dx \mathrm dy$$

but since $x,y$ are dummy variables, this is the same as:

$$\int_{\Bbb R} \int_{x - c}^\infty f(x)f(y) \ \mathrm dy \mathrm dx$$

If $c = 0$, we obtain:

$$2 G(0) = \int_{\Bbb R} \int_{-\infty}^x f(x)f(y) \ \mathrm dy \mathrm dx + \int_{\Bbb R} \int_x^\infty f(x)f(y) \ \mathrm dy \mathrm dx = \int_{\Bbb R} \int_{\Bbb R} f(x)f(y) \ \mathrm dy\mathrm dx = 1$$

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    won't it be possible to tell about the sign of the derivatives and the value of the function at $c=0$ with the information already provided? The PDF and CDF are not explicitly defined, as you mentioned they satisfy the general features of such functions.2012-10-15
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    For the derivative $G''(0)$ we can observe $f(x)f'(x) = \dfrac12 (f^2)'(x)$ to obtain that $G''(0) = \displaystyle \frac12\left(\lim_{x \to \infty} f^2(x) - f^2(-x)\right)$. It may be that you are allowed to assume this is zero.2012-10-15
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    I cannot thank you enough. I missed one point, though. I couldn't understand what exactly does "$x,y$ are dummy variables" mean and how it lead to that equivalence.2012-10-16
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    It means that replacing, say, $x$ consistently with another symbol (e.g. $\xi$) wouldn't influence the meaning of the expression. I employed it to interchange the symbols $x$ and $y$ everywhere. PS: You may thank me by accepting or upvoting the answer :).2012-10-16
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    I will be glad to do that, may you tell me how to. I am a new user of this site.2012-10-16