Here the objective is to sample from the 2-dimensional pdf

\[ f(x, y)=c \mathrm{e}^{-(x y+x+y)}, \quad x \geqslant 0, \quad y \geqslant 0 \]

for some normalization constant \(c\), using a Gibbs sampler. Let \((X, Y) \sim f\).

(a) Find the conditional pdf of \(X\) given \(Y=y\), and the conditional pdf of \(Y\) given \(X=x\).

(b) Write working Python code that implements the Gibbs sampler and outputs 1000 points that are approximately distributed according to \(f\).

(c) Describe how the normalization constant \(c\) could be estimated via Monte Carlo simulation, using random variables \(X_{1}, \ldots, X_{N}, Y_{1}, \ldots, Y_{N} \stackrel{\text { iid }}{\sim} \operatorname{Exp}(1)\).