Independent Events- Indicator Functions

Question

I can't seem to prove that two events are independent iff their indicator functions are independent discrete random variables. I was hoping to see a proof of this as I cant seem to find a proof in any of my notes nor online. Thanks

Answer 1

Two events $A$ and $B$ are independent if $$ P(A\cap B)=P(A)P(B)\tag{1} $$ and their indicator functions $\mathbf 1_A$ and $\mathbf1_B$ are independent if $$ P(\mathbf 1_A=a,\mathbf 1_B=b)=P(\mathbf 1_A=a)P(\mathbf 1_B=b)\tag{2} $$ for all choices of $a,b\in\mathbb{R}$. Then we just have to note that the sets $\{\mathbf 1_A=a\}$ when $a$ varies are actually quite simple: $$ \{\mathbf 1_A=a\}=\{\omega\mid \mathbf 1_A(\omega)=a\}= \begin{cases} A\quad &\text{if }\; a=1,\\ A^c\quad &\text{if }\; a=0,\\ \varnothing\quad&\text{if }\;a\neq 0,1. \end{cases} $$

This should enable you to quite easily show that $(2)\Longrightarrow (1)$ by choosing $a$ and $b$ wisely. To show the opposite direction, you might want to show that if $A$ and $B$ are independent, then also $A^c$ and $B$, $A$ and $B^c$, $A^c$ and $B^c$ are independent (also note that $(2)$ is obviously satisfied for $a$ and $b$ not being $0$ or $1$).