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 \lfloor x\rfloor the largest integer not exceeding x. Similarly, denote \text {frac}(x)=x-\lfloor x\rfloor to be the fractional part of x. The following are two properties of \lfloor x\rfloor and \text {frac}(x):

\displaystyle \displaystyle x \displaystyle = \displaystyle \lfloor x\rfloor +\text {frac}(x)
\displaystyle \lfloor x\rfloor \displaystyle \leq \displaystyle x<\lfloor x\rfloor +1,
\displaystyle \text {frac}(x) \displaystyle \in \displaystyle [0,1).
For example, if x=2.91, then \lfloor x\rfloor =2 and \text {frac}(x)=0.91.

Let Y=\lfloor x\rfloor and let p_ Y(y) be its PMF. There exists some nonnegative integer \ell such that p_ Y(y)>0 for every y\in \{ 0,1,\dots ,\ell \}, and p_ Y(y)=0 for y\geq \ell +1. Find \ell and p_ Y(y) for y\in \{ 0,1,\dots ,\ell \}. Your answer should be a function of k.

\ell =\quad
unanswered

p_ Y(y)=\quad
unanswered

Let Z=\text {frac}(x) and let f_ Z(z) be its PDF. There exists a real number c such that f_ Z(z)>0 for every z\in (0,c),\, and f_ Z(z)=0 for every z> c. Find c,\, and f_ Z(z) for z\in (0,c).

c=\quad
unanswered

f_ Z(z)=\quad