Properties of expectation

• Theorem: Let \(X\) be a discrete random variable; \(g\) and \(h\) functions; and \(c\) a real constant. Then, the following holds:
  • \(\mathbb{E}[g(X)] = \displaystyle{ \sum_{ x \in X(\Omega)} } g(x) \cdot p_X (x)\)
  • Linearity: \(\mathbb{E}[g(X) + h(X)] = \mathbb{E}[g(X)] + \mathbb{E}[h(X)]\)
  • Scaling: \(\mathbb{E}[cX] = c\cdot \mathbb{E}[X]\)
  • DC shift: \(\mathbb{E}[c + X] = c + \mathbb{E}[X]\)