2. Here is a way to use importance sampling to estimate a marginal density. Let fX,Y (x, y) be a…

2. Here is a way to use importance sampling to estimate a marginal density. Let fX,Y (x, y) be a bivariate density and let (X1, X2),…,(XN , YN ) ~ fX,Y . (a) Let w(x) be an arbitrary probability density function. Let f X(x) = 1 N N i=1 fX,Y (x, Yi)w(Xi) fX,Y (Xi, Yi) . Show that, for each x, f X(x) p ? fX(x). Find an expression for the variance of this estimator. (b) Let Y ~ N(0, 1) and X|Y = y ~ N(y, 1 + y2). Use the method in (a) to estimate fX(x).