Suppose I have a joint probability density function (pdf) $f\left( {r,\theta } \right)$ where, $r \in \left[ {0,\infty } \right];\theta \in \left[ {0,2\pi } \right], z\in re^{j\theta}$.
Let the pdf $f\left( {r,\theta } \right)$ is continuous and differentiable at every point over the interval mentioned above. I am trying to come up with the following relation,
$\int\limits_0^\infty {\int\limits_0^{2\pi } {{e^{r\cos (\theta - \phi )}}} f\left( {r,\theta } \right)d\theta dr} \approx \int\limits_0^\infty {g(r)f\left( r \right)dr} $
where, $f(r)$ is the marginal pdf.
- Is there any way I can use the concept of integration over the CDF or pdf (I heard about Riemann-Stieltjes integral, Lebesgue integral, etc. I am not sure if I can use this concept under some condition. )
- Is there any advice or example or theorem that can help me solve the problem.