We take the first three of the four properties of CDF proved in the previous section as the definition of a CDF or distribution function, in general.
Definition 82
A (cumulative) distribution function (or CDF for short) is any function $F:\mathbb{R}\rightarrow [0,1]$ be a non-decreasing, right continuous function such that $F(t)\rightarrow 0$ as $t\rightarrow -\infty$ and $F(t)\rightarrow 1$ as $t\rightarrow +\infty$.
If $(\Omega,p)$ is a discrete probability space and $X:\Omega\mapsto \mathbb{R}$ is any random variable, then the function $F(t)=\mathbf{P}\{\omega{\; : \;} X(\omega)\le t\}$ is a CDF, as discussed in the previous section. However, there are distribution functions that do not arise in this manner.
Example 83
Let
\[\begin{aligned}
F(t)=\begin{cases} 0 & \mbox{ if } t\le 0, \ t &\mbox{ if }0 < t < 1, \ 1 &\mbox{ if }t\ge 1. \end{cases}
\end{aligned}\]
Then it is easy to see that $F$ is a distribution function. However, it has no jumps and hence it does not arise as the CDF of any random variable on a discrete probability space.
There are two ways to rectify this issue.
- The first way is to learn the notion of uncountable probability spaces, which poses many subtleties. It requires a semester or so of real analysis and measure theory. But after that one can define random variables on uncountable probability spaces and the above example will turn out to be the CDF of some random variable on some (uncountable) probability space.
- Just regard CDFs such as in the above example as reasonable approximations to CDFs of some discrete random variables. For example, if $\Omega=\{\omega_{0},\omega_{1},\ldots ,\omega_{N}\}$ and $p(\omega_{k})=1/(N+1)$ for all $0\le k\le N$, and $X:\Omega\mapsto \mathbb{R}$ is defined by $X(\omega_{k})=k/n$, then it is easy to check that the CDF of $X$ is the function $G$ given by
\[\begin{aligned}
G(t)=\begin{cases} 0 & \mbox{ if } t\le 0, \ \frac{k}{N+1} &\mbox{ if }\frac{k-1}{N}\le t < \frac{k}{N}\mbox{ for some }k=1,2,\ldots ,N \ 1 &\mbox{ if }t\ge 1. \end{cases}
\end{aligned}\]
Now, if $N$ is very large, then the function $G$ looks approximately like the function $F$. Just as it is convenient to regard water as a continuous medium in some problems (although water is made up of molecules and is discrete at small scales), it is convenient to use the continuous function $F$ as a reasonable approximation to the step function $G$.
We shall take the second option out. Whenever we write continuous distribution functions such as in the above example, at the back of our mind we have a discrete random variable (taking a large number of closely placed values) whose CDF is approximated by our distribution function. The advantage of using continuous objects instead of discrete ones is that the powerful tools of Calculus become available to us.