<p>Parameterized quantum circuits are a key component of quantum machine learning models for regression, classification, and generative tasks. Quantum Circuit Born machines produce discrete distributions over bitstrings whose length is exactly the number of qubits. To allow for distributions on continuous variables, new models have been introduced where classical randomness is uploaded into quantum circuits and expectation values are returned with a dimensionality decoupled from qubit number. While these models have been explored experimentally, their expressivity remains underexplored. In this work, we formalize this family and establish its theoretical foundation. We prove the universality of several variational circuit architectures for generating continuous multivariate distributions and derive tight resource bounds to reach universality using tools related to the Holevo bound. Our results reveal a trade-off between the number of qubits and measurements. We further explore relaxed notions of universality and present a practical use case, outlining potential domains for quantum advantage.</p>