Are the production and decay of a resonance always independent?

The widely accepted assumption that the decay of a resonance proceeds independently of its production, quantitatively expressed as a factorizing formula for the differential cross section in the invariant mass of unpolarized resonance debris, is put under scrutiny. It is shown that the factorization is always valid for scalar and pseudoscalar resonances, although the usual version of the formula is not entirely correct. For resonances with nonzero spins the factorization does not generally take place. We deal in more detail with the spin-one case, where we show a condition on the decay matrix element that ensures the validity of the same factorizing formula as in the spinless case. This condition is satisfied for rho --> pi pi but not, e.g., for K* --> pi K or a1 --> pi rho. The formalism is applied also to the case when the resonance is produced not in two-body collisions but in the decay of a heavier particle or resonance (chain decay). Application of our formulas to the e+e- production in one-photon approximation agrees with what is known from quantum electrodynamics and thus provides another test of their soundness.