Expectation of product of more than two independent random variables 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

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:
$$
\sum_{a,b,c} abc P(A=a)P(B=b,C=c)=\sum_{a} a P(A=a)\sum_{b,c} bc P(B=b,C=c)=\mathbb E(A)\mathbb E(BC).
$$
