Every probability distribution is monotonically increasing, i.e. for a given random variable \(X\) and some real numbers \(x,y\) with \(x < y\) it follows that the probability of \(X\) taking a realization smaller or equal \(x\) is smaller or equal to the probability of \(X\) taking a realization smaller or equal \(y\), formally:
\[x < y\Longrightarrow p(X \le x)\le p(X\le y).\]
Moreover, the limits of the probability distribution, as \(x\) tends to infinity, are
\[\lim_{x\to-\infty}p(X\le x)=0\quad\quad\text{and}\quad\quad\lim_{x\to+\infty}p(X\le x)=1.\]
Proofs: 1
