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Ok, I hope this question makes sense...

Suppose the the variable y has a distribution of F(y|x).

Suppose after some work I have the following equation:

(1) $\int D dD = \int_{-\infty}^\infty[\int h(x,y,g)dg]d(F(y|x)) $

Clearly, we have that $ \int D dD = 0.5*D^2 +C_1$. Suppose that $ \int h(x,y,g)dg = g(x,y)+C_2 $

where $C_1$ & $C_2$ are both constants of integration.

My question is does $ C_2 $ necessarily depend on x and y? Put another way, can I rewrite equation (1) as:

$0.5*D^2 +C = \int_{-\infty}^\infty g(x,y)d(F(y|x)) $

or must I leave it as:

$0.5*D^2 +C_1 = \int_{-\infty}^\infty [g(x,y)+C_2]d(F(y|x)) $


Hope, I managed to make some sense here.

I do have an initial condition that I would satisfy to find the constants, its that D(x)=0 when x=0. But, I'm still confused as to what that would mean. Would that imply that we have:

$ C=\int_{-\infty}^\infty g(0,y)dF(y|x=0) $

I wouldn't worry so much about the infinite limits, its just there as a substitute for the range of y, if you prefer, you can think about the problem as being from $a$ to $b$ instead of the infinite limits.

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It seems to me that your question "does $C_2$ necessarily depend on $x$ and $y$" may indicate a misunderstanding of the concept of a constant of integration. A constant of integration can't depend on anything, since it can take an arbitrary value. An indefinite integral, as the term suggests, is not a definite quantity, with the constant of integration somehow to be determined; it represents an entire class of functions that differ only by an additive constant. It's only when you add additional conditions such as initial values that it would make sense to ask whether the constant of integration as fixed by these conditions depends on some parameters.

There's a further problem with your calculation. On the right-hand side of (1), the indefinite integral occurs inside a definite integral with infinite limits. I don't see how that could make sense, since the outer integral can exist for at most one of the functions represented by the inner integral. This might not be a problem if you intend to use the condition of the existence of the outer integral to fix the constant of integration of the inner one, but that doesn't fit with the fact that you're equating this to another indefinite integral. This equation has a class of functions on the left-hand side, and even if the right-hand side means anything, it can't mean a class of functions.

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In a multivariable (more specifically PDE) setting "constants" of integration can be functions of other independent variables. – anon Apr 18 '12 at 6:11
@anon: Yes, this is precisely the sort of setting I was referring to where there are further constraints on the constants of integration. I'm not sure that "the constants of integration are functions of other independent variables" is a good way to express this, but that's just a linguistic issue -- the main point is that I don't see anything in the question that indicates this sort of setting, and without such a setting the question whether $C_2$ depends on $x$ and $y$ makes no sense. – joriki Apr 18 '12 at 6:19
@joriki: Thanks for the input. I'm sorry if my question wasn't clear, but I think that's because I'm having trouble expressing the issue. Please see my edit to the question. – Greg Apr 18 '12 at 16:43

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