Cheatsheet

129. Cheatsheet

Non-negativity	$𝑃 (𝐴) \geq 0$
Normalization	$𝑃 (Ω) = 1$
Addition Rule (Disjoint)	$\begin{matrix} ⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖} = \emptyset \\ \Rightarrow \\ 𝑃 (⋃_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) = \sum_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖}) \end{matrix}$

Union	$⋃_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}$
Intersection	$⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}$
Empty set	$𝑃 (\emptyset) = 0$
Disjoint (Mutually Exclusive)	$𝐴 \cap 𝐵 = \emptyset$

Probability Bound	$0 \leq 𝑃 (𝐴) \leq 1$

Compliment	$𝐴^{𝑐}$
Compliment Rule	$\begin{aligned} 𝑃 (𝐴) + 𝑃 (𝐴^{𝑐}) & = 1 \\ 𝑃 (𝐴^{𝑐}) & = 1 - 𝑃 (𝐴) \end{aligned}$

Commutative Law (Union)	$𝑆 \cup 𝑇 = 𝑇 \cup 𝑆$
Associative Law (Union)	$\begin{aligned} 𝑆 \cup (𝑇 \cup 𝑈) & = (𝑆 \cup 𝑇) \cup 𝑈 \\ = 𝑆 \cup 𝑇 \cup 𝑈 \end{aligned}$
Distributive Law (Intersection over Union)	$𝑆 \cap (𝑇 \cup 𝑈) = (𝑆 \cap 𝑇) \cup (𝑆 \cap 𝑈)$
Distributive Law (Union over Intersection)	$𝑆 \cup (𝑇 \cap 𝑈) = (𝑆 \cup 𝑇) \cap (𝑆 \cup 𝑈)$
Involution (Complement of a Complement)	${(𝑆^{𝑐})}^{𝑐} = 𝑆$
Complement Law (Disjointedness)	$(𝑆 \cap 𝑆^{𝑐}) = \emptyset$
Identity Law (Union with Universal Set)	$𝑆 \cup Ω = Ω$
Identity Law (Intersection with Universal Set)	$𝑆 \cap Ω = 𝑆$



Independence	Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other

Union Bound	$𝑃 (𝐴 \cup 𝐵) \leq 𝑃 (𝐴) + 𝑃 (𝐵)$
Discrete Uniform Law	$𝑃 (𝐴) = 𝑘 \frac{1}{𝑛}$
Finite Additivity Disjoint Events	$𝑃 (⋃_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) = \sum_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖})$
Countable Additivity Disjoint Events	$𝑃 (⋃_{𝑖}^{\infty} 𝐴_{𝑖}) = \sum_{𝑖 = 1}^{\infty} 𝑃 (𝐴_{𝑖})$
De Morgan Laws	${(⋂_{𝑛} 𝑆_{𝑛})}^{𝑐} = ⋃_{𝑛} 𝑆_{𝑛}^{𝑐}$
De Morgan Laws	${(⋃_{𝑛} 𝑆_{𝑛})}^{𝑐} = ⋂_{𝑛} 𝑆_{𝑛}^{𝑐}$
Bonferroni Inequality	$𝑃 (\cap_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) \geq \sum_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖}) - (𝑛 - 1)$	$𝑃 (𝑆 \cap 𝑇) \geq 𝑃 (𝑆) + 𝑃 (𝑇) - 1$
Monotonicity	$𝐴 \subseteq 𝐵 ⟹ 𝑃 (𝐴) \leq 𝑃 (𝐵)$
Continuity of Probability	$\begin{matrix} 𝐴_{1} \subseteq 𝐴_{2} \subseteq \dots \\ \Rightarrow \\ 𝑃 (⋃_{𝑛 = 1}^{\infty} 𝐴_{𝑛}) = lim_{𝑛 \to \infty} 𝑃 (𝐴_{𝑛}) \end{matrix}$
Continuity of Probability	$\begin{matrix} 𝐴_{1} \supseteq 𝐴_{2} \supseteq \dots \\ \Rightarrow \\ 𝑃 (⋂_{𝑛 = 1}^{\infty} 𝐴_{𝑛}) = lim_{𝑛 \to \infty} 𝑃 (𝐴_{𝑛}) \end{matrix}$
Conditional probability	$𝑃 (𝐴 \| 𝐵) = \frac{𝑃 (𝐴 \cap 𝐵)}{𝑃 (𝐵)}$
Multiplication Rule (Independent)	$𝑃 (⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) = \prod_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖})$
Law of Total Probability	$𝑃 (𝐴) = \sum_{𝑖} 𝑃 (𝐴 \| 𝐵_{𝑖}) 𝑃 (𝐵_{𝑖})$
Beyes’ Theorem	$𝑃 (𝐵_{𝑗} \| 𝐴) = \frac{𝑃 (𝐴 \| 𝐵_{𝑗}) 𝑃 (𝐵_{𝑗})}{\sum_{𝑖 = 1}^{𝑛} 𝑃 (𝐴 \| 𝐵_{𝑖}) 𝑃 (𝐵_{𝑖})}$


Independence	$𝑃 (⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) = \prod_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖})$	$𝑃 (𝐴 \cap 𝐵) = 𝑃 (𝐴) \cdot 𝑃 (𝐵)$
Complement Independence	For any $𝐵_{𝑖} \in {𝐴_{𝑖}, 𝐴_{𝑖}^{𝑐}}$ , $𝑃 (⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) = \prod_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖})$	If $𝑃 (𝐴 \cap 𝐵) = 𝑃 (𝐴) 𝑃 (𝐵)$ then: $\begin{aligned} 𝑃 (𝐴 \cap 𝐵^{𝑐}) & = 𝑃 (𝐴) 𝑃 (𝐵^{𝑐}) \\ 𝑃 (𝐴^{𝑐} \cap 𝐵) & = 𝑃 (𝐴^{𝑐}) 𝑃 (𝐵) \\ 𝑃 (𝐴^{𝑐} \cap 𝐵^{𝑐}) & = 𝑃 (𝐴^{𝑐}) 𝑃 (𝐵^{𝑐}) \end{aligned}$
Conditional Independence	$𝑃 (⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖} \| 𝐶) = \prod_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖} \| 𝐶)$	$𝑃 (𝐴 \cap 𝐵 \| 𝐶) = 𝑃 (𝐴 \| 𝐶) 𝑃 (𝐵 \| 𝐶)$


Expected Value (Discrete)	$𝐸 (𝑋) = \sum_{𝑖 = 1}^{𝑘} 𝑥_{𝑖} 𝑃 (𝑋 = 𝑥_{𝑖})$
Expected Value (Continuous)	$𝐸 (𝑋) = \int_{- \infty}^{\infty} 𝑥 𝑓_{𝑋} (𝑥) 𝑑 𝑥$
Variance (Discrete)	$\begin{aligned} Var (𝑋) & = 𝐸 [{(𝑋 - 𝐸 (𝑋))}^{2}] \\ = \sum_{𝑖 = 1}^{𝑘} {(𝑥_{𝑖} - 𝐸 (𝑋))}^{2} 𝑃 (𝑋 = 𝑥_{𝑖}) \end{aligned}$
Variance (Continuous)	$\begin{aligned} Var (𝑋) & = 𝐸 [{(𝑋 - 𝐸 (𝑋))}^{2}] \\ = \int_{- \infty}^{\infty} {(𝑥 - 𝐸 (𝑋))}^{2} 𝑓_{𝑋} (𝑥) 𝑑 𝑥 \end{aligned}$


Linear Combination	$\sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝑋_{𝑖}$	$𝑎_{1} 𝑋_{1} + 𝑎_{2} 𝑋_{2} + \dots + 𝑎_{𝑛} 𝑋_{𝑛}$
Expected Value of Linear Combination	$𝐸 (\sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝑋_{𝑖}) = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝐸 (𝑋_{𝑖})$	$\begin{matrix} 𝐸 (𝑎_{1} 𝑋_{1} + 𝑥_{2} 𝑋_{2} + \dots 𝑎_{𝑛} 𝑋_{𝑛}) \\ = 𝑎_{1} 𝐸 (𝑋_{1}) + 𝑎_{2} 𝐸 (𝑋_{2}) + \dots + 𝑎_{𝑛} 𝐸 (𝑋_{𝑛}) \end{matrix}$
Variance (Independent) of Linear Combination	$Var (\sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝑋_{𝑖}) = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖}^{2} Var (𝑋_{𝑖})$
Variance (Dependent) of Linear Combination	$Var (\sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝑋_{𝑖}) = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖}^{2} Var (𝑋_{𝑖}) + 2 \sum_{1 \leq 𝑖 < 𝑗 \leq 𝑛} 𝑎_{𝑖} 𝑎_{𝑗} Cov (𝑋_{𝑖}, 𝑋_{𝑗})$