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Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense: $$P_t(y):=e^{ty}-\int_{-\infty}^\infty e^{tx}f(x,y)dx \tag{$\ast$}$$

For any $t>0$, I wish to find a root $y$ of $P_t(y)=0$ as a function of $t$ (and $f$). To wit, this looks like the stuff one sees with bilateral laplace transform resolvents or fredholm theory. My interest for now are as follows:

1) When does $P_t(y)=0$ have a solution $y(t)$, for all $t>0$?

2) When is this solution unique for each $t$?

References would be highly appreciated!

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If $f$ is compactly supported on $\mathbb{R}^2$ then for all large enough $\lvert y\rvert$, $f(x,y)=0$ for all $x$ – and so your equation cannot hold. –  Harald Hanche-Olsen Jul 4 '12 at 22:48
    
@Harold Hanche-Olsen: Pardon, I've rephrased the question for clarity. –  Alex R. Jul 5 '12 at 6:31

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