Intro to Probability - FAQ Week 4
In this space I post the most common and recurrent questions of
the week. It can be used as additional material you can use to further
and test your understanding.
Question
What is the meaning of the notations $o()$ (little o) and $O()$ (big o)?
Answer
The symbols are used to indicate order of magnitudes of
quantities. Take for instance a sequence $(x_n)_{n\in \mathbb{N}}$ of
non-negative real numbers. Then, for $k\in \mathbb{Z}$,
$$
x_n=O(n^k) \text{ as }n\to\infty
$$
means that there is a constant $C$ independent of $n\in \mathbb{N}$
such that
$$
x_n\leq C n^k\text{ for }n \text{ large.}
$$
Fromally
$$
x_n=O(n^k) \text{ as }n\to\infty\text{ iff there is} N\in \mathbb{N}
\text{ s.t. } x_n\leq C n^k\text{ for }n\geq N.
$$
This intuitively means that $x_n$ grows (decays if $k<0$) at most
(least if $k<0$) as fast as $n^k$ does when $n\to\infty$. In a similar
fashion one says that
$$
x_n=o(n^k) \text{ as }n\to\infty\text{ iff }
\lim_{n\to\infty}\frac{x_n}{n^k}=0,
$$
which means that $x_n$ grows slower (decays faster if $k<0$) than
$n^k$.
Similarly, given a function $f:\mathbb{R}\to\mathbb{R}$ and $k\in
\mathbb{N}$, we can define
\begin{align*}
f=O(h^k)\text{ as }h\to 0+&\text{ iff there is} C>0, h_0>0\text{
s.t. }|f(h)|\leq C\, h^k \text{ for }0\leq h\leq h_0\text{, and} \\
f=o(h^k)\text{ as }h\to 0+&\text{ iff }\lim_{h\to 0+}\frac{f(h)}{h^k}=0,
\end{align*}
with a similar interpretation as above. The following are examples
$$
n\log(n)=O(n^2)\text{ as }n\to\infty,\text{
}\sin(n)=O(1)=O(n^0)\text{ as }n\to\infty, \text{ }
log(n)=o(n)\text{ as }n\to\infty,\text{ }h^{1.5}=o(h)\text{ as }h\to 0+
$$
Question
Why is the cumulative distribution function of a random variable not
left-continuous in general?
Answer
It is enough to produce a counterexample. Let $S=\{ H,T\}$ be the
sample space corresponding to a fair coin toss and let the random
variable $X$ be given by
$$
X(\omega)=\begin{cases} 0,& \omega=T,\\
1,&\omega=H.\end{cases}
$$
Using the definition $F_X(x)=P(X\leq x)$ for $x\in \mathbb{R}$, it is easily seen that
$$
F_X(x)=\begin{cases} 0,& x<0,\\
0.5,&0\leq x<1,\\
1,&x\geq 1,\end{cases},
$$
which is clearly not left-continuous at $x=0$ and $x=1$.