Thread: 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(mu_x,1) and Y₁~N(mu_y,1), can be used to estimate the probability p(c)= P(XY>=c, X>=0). Discuss how mu_x and mu_y should be chosen in order for the importance sampling method to be efficeint.

Any help is much appreciated.

Hey Mullineux.

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).