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.