I want to calculate KL Divergence between a normal and an exponential r.v. i.e. $$D(P||Q) = ?\\ \;\; P=N(\mu,\sigma), \;\; Q=exp(\lambda)$$ My problem is that in this case the domains of the distributions are different - the domain of $P$ is $x\in R$ and the domain of Q is $x \in [0,\infty )$. Which domain should I integrate over? If this is the domain of $P$ the value of $\log(Q(x)/P(x))$ is not defined.
Let's say we use want to calculate the KL Divergence for $\mu = 1, \sigma = 2 ,\lambda =1$ what will be the result? I can calculate $D(Q||P)$ but it is not the same.