I have the following double integral which I would like to solve:
$$\int_{0}^{100} f_S(\sigma) \sigma \int_{-\infty}^{\infty} r e^{-\frac{(r-k\sigma)^2}{2\sigma^2}} \frac{1}{\sigma\sqrt{2\pi}} \, dr \, d\sigma$$
where k is any known parameter. Note that r is a random variable whose standard deviation, i.e., σ, is distributed lognormally. The PDF of σ is denoted in the above integral by $f_S(\sigma)$. Assume that the parameters of the lognormal distribution are known.
I thought about the following way to solve the aforementioned integral:
- The above integral may be represented as $\int_{0}^{100} \int_{-\infty}^{\infty} r\sigma f(r, \sigma) \, dr \, d\sigma$, where $f(r, \sigma)$ is a joint PDF of r and σ.
- The joint PDF of r and σ, i.e., $f(r, \sigma)$ can be found by solving $\int_{0}^{\infty} f_R(r \,|\, \sigma) K(\sigma, r) f_S(\sigma) \, d\sigma$, where $f_R(r \,|\, \sigma)=e^{-\frac{(r-k\sigma)^2}{2\sigma^2}} \frac{1}{\sigma\sqrt{2\pi}}$, $f_S(\sigma)=e^{-\frac{(ln(\sigma)-a)^2}{2b^2}} \frac{1}{\sigma b\sqrt{2\pi}}$, and $K(\sigma, r)$ is a Kernel function defined by $e^{-\frac{(r-k\sigma)^2}{2\sigma^2}}$. Recall that a and b are known.
- Finally, the numerical procedure may be applied to solve the received integral.
In this regard, I would like to ask three questions: (1) Is the way represented here is valid? If not, (2) Can you please provide the correct way, and (3) Which numerical procedure (method) may be used to solve the obtained integral numerically efficiently?
Thanks