# Importance Sampling MC - a couple of questions regarding PDF

I implore the good people to quickly glance through this thread over at StackOverflow to get a better idea of my question if the following isn't clear.

I have these integrals to evaluate using Importance Sampling--

1) x^(-0.5) ; x in [0.01,1]

2) [1+sinh(2x)ln(x)]^-1 ; x in [0.8,3]

Question 1 How does one chose the importance PDF really? For the above functions I just plotted exponential and normal functions (since it just occurred to me that they are similar to the integrals to be evaluated. Otherwise, is there a procedure to arrive at which PDF is to be used?

Question 2 Is it necessary that the chosen importance PDF has to have lesser variance than the one to be evaluated? My understanding is that the PDF chosen should follow and cover the plot of the given integral.

Question 3 So, I have a function x^2 + y^2, where x,y in[-1,1] . For cases like this, where the integral to be evaluated is a double integral, what PDF should i think about? 2-D normal? If it's a random function, there's no way to generate samples from it (for the lack of built in commands in R- only standard PDFs exist.)

• There's some basic discussion of importance sampling here Commented Feb 28, 2014 at 14:04
• One possibility for the third is to calculate it by considering it as lying inside a cube (2x2x2), and calculating the volume of the remainder; that is find $8-\int_{-1}^1\,\int_{-1}^1 2-(x^2+y^2)\, dx\; dy$. Alternatively you might consider sampling independent beta distributions with appropriately chosen parameters Commented Feb 28, 2014 at 14:39
• @Glen_b - I am reading that. Would you happen to know how to go about Question 3 ?
– Raaj
Commented Feb 28, 2014 at 14:40
• Two suggestions in my previous comment. If this is for some subject, you should add the self-study tag, and try to follow the suggested guidelines there. (You'd need to remove one of your 5 current tags first though. I think two of them aren't really needed.) Commented Feb 28, 2014 at 14:41
• Beta sort of makes sense to me. Cube doesn't . I will try to dig more into it. Thank you. If there are any links you think would be helpful, please feel free to send them my way.
– Raaj
Commented Feb 28, 2014 at 14:47