Suppose you have a smooth positive real-valued function $f$ defined on the unit cube $[0,1]^3$. Think of the unit cube as a six-sided die where, without loss of generality, the 1-side faces the positive $x$-direction, the 2-side faces the positive $y$-directions, the 3-side faces the positive $z$ direction, and opposite sides sum to 7. The function $f$ describes the density of the die at a point. What is the probability distribution of the result of rolling this die?
Context: I thought of this question while bored in a teacher meeting. It seems like a pretty natural question, but couldn't find it on this site (searching for "unfair die" or "unfair dice" returns only a pile of poorly-titled questions, in case someone has motivation to do housekeeping). Hopefully the intended question is clear. Feel free to edit it you think it needs clarification. Especially feel free to edit the tags; this is not near my area of expertise. Idk how appropriate the physics tag is.