Independence of multiple random variables.

Question

Let $\boldsymbol{X}_1,...\boldsymbol{X}_n$ and $\boldsymbol{Y}$ be random variables. Suppose that each $\boldsymbol{X}_i$ is independent of $\boldsymbol{Y}$ though $\boldsymbol{X}_i$ may not be independent of $\boldsymbol{X}_{j \neq i}$. Let $c_1,...,c_n$ be arbitrary constants. Is the random variable $\boldsymbol{Z} = \sum^n_{i=1} c_i \boldsymbol{X}_i$ independent of $\boldsymbol{Y}$?

Answer 1

Note: All operations are over $\mathbb{Z}_2$ for this answer.

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Consider random variables $\boldsymbol{X}_1, \boldsymbol{X}_2$, each of which are either $1$ or $0$ with probability $\frac 12$. Let $\boldsymbol{Y}=\boldsymbol{X}_1+\boldsymbol{X}_2$. Clearly, each pair of these three random variables is independent, but taking $c_1=1, c_2=1$ obviously causes $\boldsymbol{Z}$ and $\boldsymbol{Y}$ to be equal, and not independent. Note that this works despite $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ being independent.