Independence (Math)

Independence of Events §

thm. For partition $B_1,\dots ,B_n$ of $\Omega$, for all $A\subseteq \Omega$:

$$\begin{gathered} P(A)=P(A{B}_1)+P(A{B}_2)+\cdots +P(A{B}_n) \\ =P(A|B_1)\cdot P(B_1)+\cdots +P(A|B_n)\cdot P(B_n) \end{gathered}$$

• i.e. $P(A)$ is the weighted average of conditional probabilities $P(A|B_i)$ with weights $P(B_i)$.

def. Independence of Two Events. Two events A, B are independent if

$$\begin{gathered} P(A|B)=P(A|B^C) \\ \iff \\ P(A|B)=P(A) \end{gathered}$$

thm. Necessary and sufficient for independence of two events:

$$\begin{gathered} \text{A,B are independent} \\ \iff \\ P(AB)=P(A)\cdot P(B) \\ \iff \\ P(AB)=P(B)\cdot P(A|B) \end{gathered}$$

Warning

Events are independent not when they are related, but instead whether one event influences the probabilities of another.

def. Independence of Three Events. Three events A, B, C are independent if both:

1. They are pairwise independent
2. $P(ABC)=P(A)\cdot P(B)\cdot P(C)$

thm. Necessary and sufficient for independence of two events; if both:

1. $P(AB)=P(A)\cdot P(B)$
2. $P(C|AB)=P(C|A^{C}B)=P(C|A{B}^C)=P(C|A^C{B}^C)$ ← all of them have to be equal.

thm. Multiplication Rule for Three Independent Events.

$$P(ABC)=P(A)\cdot P(B)\cdot P(C)$$

Independence of Random Variable §

def. Independence. Two Stat 230s $X,Y$ are independent IFF for all pairs of $(x,y)$

$$\begin{gathered} {\forall }_{x,y}\mathbb{P}(X=x,Y=y)=\mathbb{P}(X=x)\cdot \mathbb{P}(Y=y) \\ \Updownarrow \\ {\forall }_{x,y}\mathbb{P}(X=x|Y=y)=\mathbb{P}(X=x) \end{gathered}$$

• i.e. the same as it is for events.

def. For $n$ random variables $X_1,\dots ,X_n$ are mutually Independent IFF for all $(k_1,\dots ,k_n)$:

$${\forall }_{x,y}\mathbb{P}(X_1=k_1,...X_n=k_n)=\mathbb{P}(X_1=k_1)\cdot \cdots \cdot \mathbb{P}(X_n=k_n)$$