Connecting with the PMF

• \(f_X(x) = \displaystyle{ \sum_{x_k \in \Omega}} p_X(x_k) \delta(x - x_k)\), where the delta functions satisfy:
  • \(\delta(x - x_k) = \infty\), if \(x = x_k\)
  • $(x - x_k) = 0, if \(x \neq x_k\)
• Rigorously, we can’t integrate such \(f_X\) proposed, but, informally, we can treat a discrete PDF as a train of delta functions.