Modeling the product of discrete * continuous

I want to model and predict a variable y which is the product of two components a discrete variable f which has a Poission distribution and s which has a positive continuous distribution. For instance if f is 0 then y is also 0. I also have some covariates x that drive f and s.

y = f * s

If I just model y directly as continuous, I don’t have componding errors but my model might also be a bit weak as it doesn’t understand the underlying semi discrete nature.

If I just model f and s, then y might be also inaccurate.

Does it make sense to have 3 likelihood equations in my model, for (f, s, y)?

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Do you have observations of f and s? If not, then I’m not sure how you could write likelihoods for them.