Following a Poisson distribution, my textbook has the following expression:
$$P(X \leq 20 | X > 15) = \frac{P(X \leq 20 | X > 15)}{P(X > 15)} = \frac{P(15 < X \leq 20)}{P(X > 15)} = \frac{P(15 < X \leq 20)}{1 - P(X \leq 15)} = \frac{F_X(20) - F_X(15)}{1 - P(X \leq 15)}$$
What I don't understand is just the lower tail in last step. Why is it not:
$$\frac{P(15 < X \leq 20)}{1 - P(X \leq 15)} = \frac{P(15 -1 \leq X \leq 20)}{1 - P(X \leq 15)} = \frac{F_X(20) - F_X(14)}{1 - P(X \leq 15)}$$
Does this also mean that in any discrete distribution the lower end should always be a strict order relation with the form $P(a < X \leq b)$ in order to apply the $F_X(b) - F_X(a)$ equivalence?
The CDF $F_X$ is defined as $F_X(x) = P((-\infty,x])$ for all $x\in\mathbb R$. The probability measure $P$ has the following property P$(B\setminus A) = P(B) - P(A)$, where the set difference $B\setminus A$ is defined as $B\cap A^\complement$.
Let $a,b\in\mathbb R$. Since $$\begin{align*} P(\{a< X \leq b\}) &= P(\{X\leq b\}\cap\{X > a\}) \\ &= P(\{X\leq b\} \cap \{X>a\}^{\complement\complement}) \\ &= P(\{X\leq b\}\setminus\{X>a\}^\complement) \\ &= P(\{X\leq b\}) - P(\{X\leq a\}) \\ &= F_X(b) - F_X(a). \end{align*}$$ Note that $\{X>a\}^\complement = \{X\leq a\}$.
In fact, this is the only rule you have to remember! Another property of probability measures is that $P(A\cup B) = P(A) + P(B)$ if $A$ and $B$ are disjoint, i. e. $A\cap B = \emptyset$. Now note that $$P(\{a\leq X\leq b\}) = P(\{a<X\leq b\}) + P(\{X=a\})$$ as $\{a<X\leq b\}\cap \{X=a\} = \emptyset$, and $$P(\{a < X < b\}) = P(\{a<X\leq b\}) - P(\{X=b\}).$$ Simply add the zero $P(\{X=b\}) - P(\{X=b\})$ to $P(\{a<X<b\})$ and use the fact that $\{a<X<b\}\cap\{X=b\} = \emptyset$.
Both cases can be of course combined to compute $P(\{a\leq X<b\})$: $$P(\{a\leq X<b\}) = P(\{a<X\leq b\}) + P(\{X=a\}) - P(\{X=b\}).$$
The general rule is thus $$P(\{a\star X\bullet b\}) = F_X(b) - F_X(a) + \alpha P(\{X=a\}) - \beta P(\{X=b\}),$$ where $\alpha = 1$ if $\star = \leq$ and $\alpha=0$ otherwise, and $\beta = 1$ if $\bullet = <$ and $\beta=0$ otherwise.