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I have $X_1,X_2,X_3$ from $N(1,2)$ distribution. Let's define random variables $$Y_1=X_1+X_2,\; Z_1=X_3-X_2+X_1$$. I want to prove or disprove idenepndence of random variables $Z_1$ and $Y_1$.

My work so far

Firsly we see that $Y_1$ has a $N(2,\sqrt{2^2+2^2)}=N(2,2\sqrt{2})$ distribution and $Z_1$ has $N(2,2\sqrt3)$ distirubtion.

If they were independent then the following equality has to be true : $$P(Y_1 \in [a_1,b_1],Z_1 \in [a_2,b_2])=P(Y_1\in [a_1,b_1])P(Z_1\in[a_2,b_2])$$

And we have

$P(Y_1\in[a_1,b_1])=F_{2,2\sqrt{2}}(b_1)-F_{2,2\sqrt{2}}(a_1)$

$P(Z_1\in[a_2,b_2])=F_{2,2\sqrt{3}}(b_2)-F_{2,2\sqrt{3}}(a_2)$

And I have little troubles with telling what's the probability in the left side of the equality. I tried something like $P(X_1 \le min(b_1-X_2,b_2+X_3-X_2))$ (for upper bound but I dont think it's a good idea.

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I assume that $X_1$, $X_2$, and $X_3$ are i.i.d. Since $Y_1$ and $Z_1$ are jointly normal it suffices to compute their covariance. Specifically, $Y_1$ and $Z_1$ are indpendent iff $\operatorname{Cov}(Y_1,Z_i)=0$. Let $X_j':=X_j-1$. Then $$ \operatorname{Cov}(Y_1,Z_i)=\mathsf{E}(X_1'+X_2')(X_3'-X_2'+X_1')=\operatorname{Var}(X_1)-\operatorname{Var}(X_2)=0. $$

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I suppose $X_i$'s are assume to be independent.

One way to answer this question is to see if $Ee^{itY_1+is Z_1}=Ee^{itY_1} Ee^{itZ_1}$ for all $s$ and $t$. My calculation shows that both sides are equal to $e^{i(2t+s)} e^{-2t^{2}} e^{-3s^{2}}$. So $Y_1$ and $Z_1$ are independent.

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