I have some truncated normal distributions and some other distributions (like uniform distribution) that have partially common domains. I'd like to know how can I calculate the entropy for the sum of these distributions. Let X~Uniform(a,b) and Y~truncatedNormal(c,d) and Z~uniform(e,f) and a < c < e < b < d < f. Now I want to calculate the entropy of the distribution of the sum.
Independence means exactly that the joint pdf is the product of the individual pdfs.
If we are talking about probability theory, and the densities are not independent, then you have to specify how they depend on each other.
About entropy, an easy way of seeing it is through conditional entropy: the conditional entropy $H(X|Y)$ is the extra amount of entropy that you get from $X$ by knowing $Y$. Then you can write $$ H(X,Y) = H(X|Y)+H(Y) $$
of course, if $X$ depends completely from $Y$, then $H(X|Y)=0$. If instead the information of $X$ that you get from $Y$ is null, therefore $X$ and $Y$ are independent, then $H(X|Y)=X$. In this way, you can quantify the joint entropy from the individual distributions and their dependence.
The conditional entropy is computed as $$ H(X|Y)=\int dy\;p(y)\;H(X|Y=y) = -\int dy\;p(y) \int dx\;p(x|y)\;log\left(p(x|y)\right) $$
so the most convenient way of writing the dependence of the two variables would be through their conditional density, $p(x|y)$.