Maybe I'm not understanding these concepts correctly, but it seems like the idea of a mixture model seems to be superfluous, since we can express any mixture of random variables as a multivariate distribution, and marginalize it to get the exact same thing as the mixture.
I'm using the definitions below: https://en.wikipedia.org/wiki/Mixture_distribution https://en.wikipedia.org/wiki/Marginal_distribution
In particular, if we toss two different coins, one 90% of the time and another 10% of the time, where the first is a fair coin and the second is heads 70%, is this not modeled by a multivariate distribution, the two variables being coin type and the other being Heads/Tails in this way:
| H | T | |
|---|---|---|
| Coin 1 | .45 | .45 |
| Coin 2 | .07 | .03 |
And you can just take the marginal distribution for H-T from there?
In your case you do not need this. If you know the weights for each category and the distriution of each category, you can trivially calculate the total distribution. But what if you do not know how fair your two coins are? This is what a model is. We do not know some parameters, and we try to find these parameters from observations.