# Independence of random variables and events

I was reading text on probability (in discrete mathematics) and there they state that:

A set of events $A_1, A_2, \dots A_n$ is mutually independent iff for $\forall S \subset [1,n]$: $Pr(\cap_{j\in S}A_j)=\prod_{j\in S} Pr(A_j)$.

Thus to prove that a set of events is independent we need to consider all the subsets of the set.

And when they talked about probability of random variables they stated that:

Random variables $R_1, R_2, \dots R_n$ are mutually independent iff $Pr(R_1=x_1\cap R_2=x_2\dots R_n=x_n)=\prod_{1\leq i\leq n}Pr(R_i=x_i)$ for all $x_1,x_2,\dots x_n$.

What I understood is that random variable taking a certain value is in itself an event, however still when checking for mutual independence among random variables, we are not considering all the subsets of the random variables as we did in case of events but we check for all values of $x_i$.

Can anyone please explain me why we are doing so?

Events $A_1,\dots,A_n$ are (also) mutually independent if and only if $$P(E_1\cap\cdots\cap E_n)=P(E_1)\times\cdots\times P(E_n)$$

For all tuples $(E_1,\dots,E_n)$ where $E_i\in\{A_i,A_i^{\complement}\}$.

This principle is applied in the case of the discrete random variables, so makes it unnecessary to look at subsets of the index-set.

Note that from what you mention as definition of independence for discrete random variables it can be deduced that:$$P(R_1\in B_1\cap\cdots\cap R_n\in B_n)=\prod_{i=1}^nP(R_i\in B_i)$$ for sets $B_i$.

Then applying what is mentioned above we find that the events $\{R_1\in B_1\},\dots,\{R_n\in B_n\}$ are mutually independent.

• Can you mention some link where I could read about the above definition you provided for independence of events as I can't find it anywhere. Thanks. – shiva Jun 11 '18 at 18:56
• I think I got it, in a way we are converting events to their independent random variables, and using their definition of independence. Am I right? – shiva Jun 11 '18 at 18:59
• Yes. The events $A_i$ are independent if and only if the corresponding indicator functions $1_{A_i}$ are independent random variables. – drhab Jun 11 '18 at 20:05