Integer and Fractional Parts of a Uniform Random Variable Let k be a positive integer and let X be a continuous random variable that is uniformly distributed on [0, k]. For any number , denote by [x] the largest integer not exceeding a. Similarly, denote frac (x) = x[x] to be the fractional part of x. The following are two properties of [x] and frac(x): = [x]+frac(x) 2 1 0 for every y = {0,1,..., }, and Py (y) = 0 for y>l+1. Find land py (y) for y = {0,1,...,l}. Your answer should be a function of k. l = PY (y) = 2. Let Z= frac(x) and let fz (2) be its PDF. There exists a real number c such that fz (2) > 0 for every z = (0,c), and fz (z) = 0 for every z > c. Find c, and fz (2) for z = (0,c). c= fz (2)=