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: $\mathbb{E}[c + X] = c + \mathbb{E}[X]$