A Characterization of Uniform Distribution

Volume 53 / 2005

Joanna Chachulska Bulletin Polish Acad. Sci. Math. 53 (2005), 207-220 MSC: 60E05, 62E10, 28A12. DOI: 10.4064/ba53-2-9


Is the Lebesgue measure on $[0,1]^2$ a unique product measure on $[0,1]^2$ which is transformed again into a product measure on $[0,1]^2$ by the mapping $\psi(x,y)=(x,(x+y)\bmod 1))$? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables $X$ and $Y$ constancy of the conditional expectations of $X+Y-I(X+Y>1)$ and its square given $X$ identifies uniform distribution either absolutely continuous or discrete. No assumptions are imposed on the supports of the distributions of $X$ and $Y$.


  • Joanna ChachulskaFaculty of Mathematics and Information Science
    Warsaw University of Technology
    Pl. Politechniki 1
    00-661 Warszawa, Poland

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