# Discrete random variable and probability mass function

Summary

- A (real-valued) random variable \(X\) is said to be discrete if the range of \(X\) is finite or countably infinite.
- If \(X\) is a discrete random variable then the probabillity mass function (pmf) \(f: R→[0,1]\) is defined as

\[f(x)=P(X=x)\]

If \(X\) is a discrete random variables that can take values \(x_1,x_2...,x_n\) , then the probability mass function of \(X\) must satisfy the following conditions

- \(f\left( x_i \right)≥0\) for \(i=1,2,…,n\)
- \(\sum_{i=1}^{n}{ }f(x_i)=1\)

If the range is countably infinite you replace \(n\) with \(∞\) .