Probability Mass Function

A Discrete real-valued function that maps possible values of Random Variable into $[0,1]$ probabilities

Notation

$p_X(x_i)$ This mapping is often not represented as a formula, but a table instead.

• For Discrete Distribution, $p_X(x_i)\equiv P(X=x_i)=P(\{s\in S|X(s)=x_i\})$

Axioms

• $0\le p_X(x_i)\le 1,\forall i$
• ${\sum }_{\forall i}{p}_X(x_i)=1$

Example

$$\{\begin{array}{ll}\frac{1}{36}& x=2\\ \frac{2}{36}& x=3\\ \frac{3}{36}& x=4\\ \frac{4}{36}& x=5\\ \frac{5}{36}& x=6\\ \frac{6}{36}& x=7\\ \frac{5}{36}& x=8\\ \frac{4}{36}& x=9\\ \frac{3}{36}& x=10\\ \frac{2}{36}& x=11\\ \frac{1}{36}& x=12\\ 0& x<2\vee x>12\end{array}$$

It can alternatively be expressed as:

$$\{\begin{array}{ll}\frac{6-|x-7|}{36}& ,x\in [2,12]\\ 0& ,x\notin [2,12]\end{array}$$