Exercise 3.4

To show that $F(x)$ 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 $\underset{x\to a^{\text{+}}}{\lim <!--\; FUNCTION\; APPLICATION\; -->}F(x)=F(a)$ when $a\in \mathbb{N}$ by the definition of $F(x)$, the second condition is satisfied.

$\underset{x\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}F(x)=1$ by the definition of $F(x)$, and also, by definition, $\underset{x\to -\infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}F(x)=0$. Thus, the third condition is satisfied, and $F(x)$ is a CDF.

The PMF $F$ corresponds to is

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