It is shown that a lattice distribution defined on a set of n lattice points $L(n,delta) = {delta,delta+1,...,delta+n-1}$ is a distribution induced from the distribution of convolution of independently and identically distributed (i.i.d.) uniform [0,1] random variables. Also the m-th moment of the lattice distribution is obtained in a quite different approach from Park and Chung (1978). It is verified that the distribution of the sum of n i.i.d. uniform [0,1] random variables is completely determined by the lattice distribution on $L(n,delta)$ and the uniform distribution on [0,1]. The factorial mement generating function, factorial moments, and moments are also obtained.