# Find the value of an integral using Monte-Carlo method

I have a task in a subject called "Monte-Carlo methods" with which I'm a bit stuck and therefore I'm asking for your help.

The task is as follows: Describe in detail one specific option how to find approximately a value of the integral using Monte-Carlo methods and using ONLY independent random variables from uniform distribution with parametres (0,1):

$$\int_{2}^{4}(\int_{1}^{\infty} e^{x+y-xy} dx) dy$$

What I've thought so far is that Importance Sampling Method might be a good idea to solve it. So I would choose a suitable density functions for X ja Y. For X it would be a uniform distrbution with parametres (2,4) since it's values are bounded and for Y I would choose exponential distrbution with rate 1 (Exp(1)).

But the main problem relies in the second part of the question - how to use ONLY values from uniform distribution (0,1). I know that I have to use the inverse of a density function, but I'm not able to write it down as requested.

I hope you understand my question and can help me with this one!

• I suspect the (easily demonstrated) fact that the log of a uniformly distributed random variable has an exponential distribution might be very useful for this problem. – whuber Jun 16 '18 at 12:07