The outcome of a [[Y1Q2/Simulating and Modelling to Understand Change/Module I - Introduction & Random Variables Simulation/Random Variable Simulation/Discrete Random Variables | discrete random variable]] is, in general unknown, but we want to associate a probability to each element of $X$, denoted by $P$.
The pmf (probability mass function) of a random variable $X$ with sample space $X$ is defined as
$$ p(x)=P(X=x), \text{ for all } x \in X $$Pmfs must obey two conditions:
- $p(x) \geq 0$, for all $x \in X$
- $\sum_{x \in X} p(x) = 1$
The pmf associated to each outcome is a non-negative number such that the sum of all these numbers is equal to one.
Take the example of a biased dice, such that 3 and 6 are twice as likely to appear than other numbers. Pmf:
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| p(x) | 1/8 | 1/8 | 2/8 | 1/8 | 1/8 | 2/8 |
|
Probability Density Function