# Solve Joint Distribution From An Equation

In Postel-Vinay & Robin 2002 they show an equation: $$\left\{\delta+\mu+\lambda_{1} \bar{F}(p)\right\} \ell(\varepsilon, p)=\left\{(\delta+\mu) h(\varepsilon)+\lambda_{1} \int_{p_{\min }}^{p} \ell(\varepsilon, x) d x\right\} f(p)$$ where $$h$$ is the pdf of $$ε$$, and $$\bar{F}(p)=1-F(p)$$ is the inverse cdf of $$p$$.

They then say that this solves as $$\ell(\varepsilon, p)=\frac{1+\kappa_{1}}{\left[1+\kappa_{1} \bar{F}(p)\right]^{2}} h(\varepsilon) f(p)$$ where $$\kappa_{1}=\lambda_{1} /(\delta+\mu)$$, but I can't see how.

Also note that at this time we don't know that if the joint distribution is independent or not because the authors integrate this result to get $$\ell(p)=\frac{1+\kappa_{1}}{\left[1+\kappa_{1} \bar{F}(p)\right]^{2}} f(p)$$ and thus $$\ell(\varepsilon, p)=h(\varepsilon) \ell(p)$$ and finally argue that $$ε$$ and $$p$$ are uncorrelated.