Question

Let k be a positive integer and let X be a continuous random variable that is uniformly distributed on [0,k]. For any number x , denote by ⌊x⌋ the largest integer not exceeding x. Similarly, denote frac(x)=x − ⌊x⌋ to be the fractional part of x. The following are two properties of ⌊x⌋ and frac(x):
x =⌊x⌋+frac(x)
⌊x⌋ ≤ x < ⌊x⌋+1
frac(x) ∈[0,1)
For example, if x=2.91, then ⌊x⌋=2 and frac(x)=0.91.
1. Let Y=⌊x⌋ and let pY(y) be its PMF. There exists some nonnegative integer ℓ such that pY(y) > 0 for every y∈{0,1,…,ℓ}, and pY(y) = 0 for y ≥ ℓ+1
Find ℓ and 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(z) be its PDF. There exists a real number c such that fZ(z) > 0 for every z ∈ (0,c), and fZ(z) = 0 for every z > c. Find c, and fZ(z) for z∈(0,c).

c =
fZ(z) =