On independence of time and cause

For two independent, almost surely finite random variables, independence of their minimum (time) and the events that either one of them is equal to the minimum (cause) is completely characterized. It is shown that, other than for trivial cases where, almost surely, either one random variable is strictly greater than the other or one is a constant and the other is greater than or equal to it, this happens if and only if both random variables are distributed like the same strictly increasing function of two independent random variables, where either both are exponentially distributed or both are geometrically distributed. This is then generalized to the multivariate case.