I would appreciate help showing (all the way) that the mixed variable Z=AX+BY while X,Y~N(0,1) and...

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I would appreciate help showing (all the way) that the mixed variable Z=AX+BY while X,Y~N(0,1) and A,B can get -1 or 1 in the same probability (0.5) is normal distributed variable and finding its parameters, meaning its mean and variance I would appreciate help showing (all the way) that the mixed variable Z=AX+BY while X,Y~N(0,1) and A,B can get -1 or 1 in the same probability (0.5) is normal distributed variable and finding its parameters, meaning its mean and variance