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I present a theory of random consumer demand. The primitive is a collection of probability distributions on budgets. Axioms constrain these distributions, including analogues of preference axioms, such as transitivity, monotonicity and convexity. Results establish a complete representation of theoretically consistent demand. The theory's purpose is empirical application. To this end, the theory has desirable properties. Intrinsically stochastic, econometricians can apply it without adding extrinsic randomness in the form of errors. Random demand is parsimoniously represented by a single function on the consumption set. Finally, there exist practical methods for inference based on the theory, described in a companion paper.