Probability Mass Function

Probability mass functions are used to define the distributions of discrete random variables $X:\Omega \to \mathcal{X}$ with discrete and finite (or least countable) outcome set $\mathcal{X}$.

Statements

Definition. Let $(\Omega ,\mathcal{F},P)$ be a probability space. A random variable $X:\Omega \to \mathcal{X}$ is said to be discrete, if its outcome space $\mathcal{X}$ contains countably many elements $x_i,i=1,2,\dots$. The probability mass function (PMF) of a discrete random variable $X$ is denoted by $p_X$ and is defined as

$$p_X:\mathcal{X}\to [0,1]$$ $$p_X(x_i)=P_X(X=x_i)$$

References

D. Ostwald - The general linear model 20_21

Probability mass function - Wikipedia