I am trying to use Monte Carlo method to solve problems regarding double integrals. In particular, the integral to be solved is in the form of $$ \int_{b}^{\infty}\int_{a}^{\infty} f(t|x)g(x) dx dt. $$
Here, $f(t|x)$ is the conditional probability where $$f(t|x)=e^{\beta x}\lambda e^{-e^{\beta x}\lambda t}, g(x)=\mu e^{-\mu x}.$$
I am aware the existence of alternatives numerical procedure such as cubature and quadrature. However, I'm prone to use Monte Carlo method as it is easier for coding implementation. To deal with the improper integrals, change of variable is needed.
Let $x=a+\frac{u}{1-u}$ and $t=b+\frac{v}{1-v}$, then the integrands are updated as $$ \int_{0}^{1}\int_{0}^{1}f\bigg(b+\frac{v}{1-v}\bigg|a+\frac{u}{1-u}\bigg)g\bigg(a+\frac{u}{1-u}\bigg) \times \frac{1}{(1-u)^2(1-v)^2} dudv. $$
Next, we can use the standard procedure of Monte Carlo method which is to generate $u$ and $v$ from uniform distribution and approximate the integration above as $$ I\approx (1-0)(1-0) \frac{\sum_{i=1}^{N} \frac{f\big(b+\frac{v_{i}}{1-v_{i}}\big|a+\frac{u_{i}}{1-u_{i}}\big)g\big(a+\frac{u_{i}}{1-u_{i}}\big)}{(1-u_{i})^2(1-v_{i})^2}}{N} $$ Can I know whether this method is valid?
Since the $t$ is dependent on $x$ and I'm not sure whether we can simply do the change of variable as above or not.