Big O Magnitude Notation

Like Big O Unit Notation, Big O Magnitude Notation refers to a notation used to express orders of magnitude.

Formal Definition for Functions

A function $f:\mathbb{ℝ}\to \mathbb{ℝ}$ is considered to be $f(n)=O(10^k)$ for some fixed $k\in \mathbb{\mathbb{Z}}\mathbb{+}$ if $\mathrm{\exists }{n}_0\in \mathbb{ℝ}\mathbb{+}$ such that $\mathrm{\forall }n\ge n_0:10^k\le f(n)<10^{k+1}$

Formal Definition for Random Variables

A random variable $X=O(10^k)$ if $P[10^k\le X<10^{k+1}]=1$.