Mass and Density Functions

TODO: Connection between PMF/PDF and CDFs

Cumulative Distribution Functions (CDFs)

These exist for both discrete and continuous Random Variables. They are immensely useful since, in the case of Random Variables with continuous distributions, you just get the probability $P(X \leq x)$ by using the PDF.

Here are some other properties:

1. If $a \lt b$ then $F_X(a) \leq F_X(b)$
   It is non-decreasing. Note the $\leq$

2. $\lim_{x \to +\infty} F_X(x) = 1$ and $lim_{x \rarr -\infty}F_X(x) = 0$
   This is more important than it looks! Why? You’d be breaking some probability axioms otherwise!

3. $P(X \gt x) = 1 - F_X(x)$
   Called “right continuity”

4. $a \lt b \implies P(a \lt X \leq b) = F_X(b) - F_X(a)$
   Easier than integrating a PDF!

Media

• A nice video on PDFs, PMFs, CDFs by ZedStatistics.
• Grant Sanderson helpfully explains that $\int{f(x)}dx$ is not a ‘real’ sum and that it becomes the ‘real’ sum $\Sigma{f(x)}$ when $dx\to\infty$. Which makes sense.