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?
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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.
The variance of the sum of two independent random variables is the sum of their variances, as is the variance of the difference of two independent random variables, so