Expectation of product of more than two independent random variables

Question

I am trying to determine whether independence of random variables changes when multiplied with other, potentially dependent variables. The question isn't in a measure-theoretic context.

In particular, I have 3 random variables, which we can call A, B, C

A is independent of both B and C, but B and C are not independent. Is it true then that:

$$\mathbb{E}[ABC] = \mathbb{E}[A]\mathbb{E}[BC]?$$

If it's true, I would appreciate if someone could either show me a proof (preferably one for someone without a strong measure-theoretic probability background) or point me to one.

Thanks

Answer 1

Let's assume you have discrete random variables so that we do not deal with integrals. First use the independence assumption: $$ \mathbb E(ABC)=\sum_{a,b,c} abc P(A=a,B=b,C=c)=\sum_{a,b,c} abc P(A=a)P(B=b,C=c). $$ And then use the following identity to get the result: $$ \sum_{a,b,c} abc P(A=a)P(B=b,C=c)=\sum_{a} a P(A=a)\sum_{b,c} bc P(B=b,C=c)=\mathbb E(A)\mathbb E(BC). $$