3.1.4 problem 4

To show that $F\left(x\right)$ is a CDF, we need to show that $F$ is increasing, right-continuous, and converges to $0$ and $1$ in the limits.

The first condition is true since $\lfloor x\rfloor$ is increasing.

Since $\underset{x\to a^{+}}{\lim }F\left(x\right)=F\left(a\right)$ when $a\in N$ by the definition of $F\left(x\right)$, the second condition is satisfied.

$\underset{x\to \infty }{\lim }F\left(x\right)=1$ by the definition of $F\left(x\right)$, and also, by definition, $\underset{x\to -\infty }{\lim }F\left(x\right)=0$. Thus, the third condition is satisfied, and $F\left(x\right)$ is a CDF.

The PMF $F$ corresponds to is

for $1\le k\le n$ and $0$ everywhere else.