We consider the set Mn of all n-truncated power moment sequences of probability measures on [0,1]. We endow this set with the uniform probability. Picking randomly a point in Mn, we show that the upper canonical measure associated with this point satisfies a large deviation principle. Moderate deviation are also studied completing earlier results on asymptotic normality given by Chang, Kemperman and Studden [Ann. Probab. 21 (1993) 1295–1309]. Surprisingly, our large deviations results allow us to compute explicitly the (n+1)th moment range size of the set of all probability measures having the same n first moments. The main tool to obtain these results is the representation of Mn on canonical moments [see the book of Dette and Studden].