# For System dependent on normally distributed parameter, are deviations added or variations?

Say, A and B are two normally distributed parameters with their variations being $\sigma^2_a$ and $\sigma^2_b$. Now for system C, which is linearly dependent on these parameters, is its $\sigma^2_c=\sigma^2_a+\sigma^2_b$, or $\sigma_c=\sigma_a+\sigma_b$. To me adding parameter deviations seems natural. But what is the actual behaviour (and why)?

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If A and B are normally distributed random variables that are independent and have variances σ$^2$a and σ$^2$b respectively and both have mean 0 then if C=A+B, C will be normally distributed with mean 0 and variance σ$^2$a+σ$^2$b. So σc=√(σ$^2$a+σ$^2$b) and not σa + σb. The reason is that for independent random variables the variance of the sum is the sum of the variances.