We compute the expected degree of a randomly chosen element in a basis of weight vectors of an arbitrary Demazure module of sl2^ by induction along Demazure�s character formula. Along those lines we obtain a new proof of Sanderson�s dimension formula for these Demazure modules. Furthermore, we compute the covariance of the full weight distribution in level 1 Demazure modules of sl2^. The crucial step is to compute the variance of the degree distribution. The knowledge of the covariance allows us to prove the weak law of large numbers for the degree and full weight distribution using Chebyshev�s inequality. We give two proofs of our results concerning level 1 Demazure modules, one by induction along Demazure�s character formula, and one by using quantum calculus and the fact that the characters of level 1 Demazure modules are related to Macdonald and Rogers�Szegö polynomials
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