Discrete random variable and probability mass function


  • 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


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 \(∞\) .