Importance Sampling Problem
I'm currently stuck on a problem that I can't seem to get my head around. Here it is:
Let X,Y~N(0,1) be independent.
Explain how importnace sampling, with samples from X1~N(mux,1) and Y1~N(muy,1), can be used to estimate the probability p(c)= P(XY>=c, X>=0). Discuss how mux and muy should be chosen in order for the importance sampling method to be efficeint.
Any help is much appreciated.
Re: Importance Sampling Problem
In terms of the technique, consider an indicator variable for Z = XY if XY >= c and then find E[Z;X>=0].
You can use the fact that the probability will correspond to a Bernoulli distribution where if I[Z] = 0 if XY < c and 1 if XY >=c then E[I[Z]] = P(XY >= c).