# Proving independence of two i.i.d random variables from standard normal distribution

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.

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 $$\operatorname{Cov}(Y_1,Z_i)=\mathsf{E}(X_1'+X_2')(X_3'-X_2'+X_1')=\operatorname{Var}(X_1)-\operatorname{Var}(X_2)=0.$$
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.