What's the difference between add and subtract of two random variables, if zero mean?

Assume $X$, $Y$ are independent zero mean random variables. Define $Z_1=X+Y$ and $Z_2=X-Y$. Then, their mean values are the same.

How does one check that $Z_1$ and $Z_2$ are not the same random variables? And how to describe their variances?

• Your question lacks some context. Supplying it might help attract an answer. As it stands, it's not clear what your question really is. As currently posed, the answer is that $Z_1 = Z_2$ (almost surely) if and only if $Y = 0$ almost surely. – cardinal Jun 18 '11 at 15:48

The difference between $Z_1$ and $Z_2$ is $2Y$. Whenever $Y \not = 0$, they will be different. If $Y$ has a symmetric distribution about $0$ then $Z_1$ and $Z_2$ will have the same distribution as each other.
$$\sigma_{Z_1}^2=\sigma_{Z_2}^2=\sigma_{X}^2+\sigma_{Y}^2$$
• Thanks. I have understand the calculation of variance.But if Y!=0, how to explain $Z_1!=Z_2$? – sam Jun 19 '11 at 7:52
• @sam: No - in your example $Cov(X,Z_1) = Var(X)$ even though they are different. – Henry Jun 19 '11 at 17:32