Suppose $X_1,X_2,\ldots,X_n$ are iid random variables with mean $0$. and let $Y_i=\sum_{k=1}^nX_k -X_i$. Clearly $X_i$ and $Y_i$ are independent. I want to find the expectation for $i \neq j$
$$\mathbb E[X_iX_jf(Y_i)]$$
Since $X_j$ and $Y_i$ are both independent of $X_i$ I was thinking that $$\mathbb E[X_iX_jf(Y_i)]=\mathbb E[X_i] \mathbb E[f(Y_i)X_j]=0$$ But I came across this question which says that for three random variables $X,Y,Z$ given $X$ is independent to $Y$ and $Z$, does not imply that $X$ is independent to $YZ$. So I guess my arguement is wrong. Is there any way to calculate the expectation of the product?