Solve the following problem:
Monte Carlo marginalization is a technique for calculating a marginal density when simulating from a joint density. Let (Xi, Yi) ∼ fXY (x, y), independent, and the corresponding marginal distribution fX(x) = R fXY (x, y)dy.
a. Let w(x) be an arbitrary density. Show that
![384_equ.jpg](https://secure.tutorsglobe.com/CMSImages/384_equ.jpg)
which provides a Monte Carlo estimate of fX, the marginal distribution of X, when the joint distribution is only known up to a constant.
b. Let X|Y = y ∼ G(y, 1) and Y ∼ Exp(1). Use the technique above to plot the marginal density of X. Compare this with the exact marginal.
c. Show that choosing w(x) = fX(x) works to produce the marginal distribution and that it is optimal in the sense of the variance of the resulting estimator.