# Independence of 4 random variables

$$A_1$$, $$A_2$$, $$B_1$$ and $$B_2$$ are random variables with

$$P(A_1, A_2, B_1, B_2) = P(A_1, A_2) P(B_1, B_2)$$.

Are $$A_1$$ and $$B_1$$ independent? I would think not, because if e.g. $$P(A_2)=0$$ and $$P(B_2)=0$$, then the equation above would be correct for dependent $$A_1$$ and $$B_1$$.

Thanks for any help and thoughts!

• You probably mean events and not random variables. In that case your thinking is correct. – Kavi Rama Murthy Nov 15 '18 at 12:07

As it is said in the comment, be careful not to mix random variables with events : $$P(A_1)=0$$ means nothing for a random variable. An event is something that might happen or not and it has a probability while a random variable is a variable that has a random value and it has not a single probability.

Events and random variables are related : you can create a random variable from an event by saying that its value is 1 when then event happens and 0 otherwise. Reciprocally, for any random variable $$V$$, you can consider one event for each of its possible values : $$V=1$$, $$V=2$$,...

Your equation (E) : $$P(A_1, A_2, B_1, B_2) = P(A_1, A_2) P(B_1, B_2)$$ is the definition of independence between the unions of variables $$\{A_1,A_2\}$$ and $$\{B_1,B_2\}$$.

But its arguments are random variables and it is a convenient notation that actually means one equation for each combinations of the related events.

eg : $$P(A_1=T,A_2=F,B_1=1,B_2=F)=P(A_1=T,A_2=F)P(B_1=T,B_2=F)$$

If your random variables are associated with 4 events, (E) is actually 16 equations, not only the one associated with True values.

So if all those 16 equations are true, you have independence of $$\{A_1,A_2\}$$ and $$\{B_1,B_2\}$$ that implies the independence between any subsets and so independence between $$A_1$$ and $$B_1$$.