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In case $X_1$ and $X_2$ are i.i.d random variables and $Y_1$ and $Y_2$ are also i.i.d random variables. Is it the case that $Z_1 = X_1 + Y_1$ and $Z_2 = X_2 + Y_2$ are independent?

Edit: I tried to figure it out for continuous variables by demonstrating $f_{Z_1 ,Z_2 } = f_{Z_1}\cdot f_{Z_2} $. My second try was to show the $P(Z_1|Z_2)=P(Z_1)$ but I failed either. Also my intuition is not so strong about it cause in case $X_1$ and $Y_2$ depends in each other for example it might lead to dependency between $Z_1$ and $Z_2$ .

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    $\begingroup$ What have you tried? Please tell us. In particular: What is the definition of independence and how would you use it? $\endgroup$ Commented Sep 3, 2017 at 6:00

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Also my intuition is not so strong about it cause in case $X_1$ and $Y_2$ depends in each other for example it might lead to dependency between $Z_1$ and $Z_2$ .

Actually, your intuition is pretty good.

Suppose $Y_1:=X_2, Y_2:=X_1$ where $X_1,X_2$ are iid.   Then $Y_1$ and $Y_2$ are also iid, but what of $X_1+Y_1$ and $X_2+Y_2$?   …

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