PX284 - A1 - discrete probability distributions

suppose $x$ is a discrete random variable with values, $x_{i}$ , and probabilities of taking the values, $P_{i}$

\sum_{i} P_{i} = 1

\begin{aligned} ⟨ x ⟩ & = \sum_{i} x_{i} P_{i} \\ ⟨ f (x) ⟩ & = \sum_{i} f (x_{i}) P_{i} \end{aligned}

σ_{x}^{2} = ⟨ (x_{i} - ⟨ x ⟩)^{2} ⟩ = ⟨ x^{2} ⟩ - ⟨ x ⟩^{2}

σ_{x} = \sqrt{⟨ x^{2} ⟩ - ⟨ x ⟩^{2}}