Discrete probability distribution

Discrete probability distribution

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values (almost surely) which means that the probability of any event ${\displaystyle E}$ can be expressed as a (finite or countably infinite) sum: $${\displaystyle P(X\in E)=\sum _{\omega \in A\cap E}P(X=\omega ),}$$ where ${\displaystyle A}$ is a countable set with ${\displaystyle P(X\in A)=1}$. Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a probability mass function${\displaystyle p(x)=P(X=x)}$. In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if ${\displaystyle p(n)={\tfrac {1}{2^{n}}}}$ for ${\displaystyle n=1,2,...}$, the sum of probabilities would be ${\displaystyle 1/2+1/4+1/8+\dots =1}$.