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I'm curious, when calculating the sum of two distribution, why do we add variance of two distribution but can't add standard deviation of two distribution?

Does it have something to do with the pythagorean theorem?

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$$a^2=b^2+c^2$$ $$\implies a=\sqrt{b^2+c^2}\neq b+c$$

Just replace $b^2$ and $c^2$ with the respective variances: $\sigma_X^2$ and $\sigma_Y^2$, and with $a^2$ being $\sigma_{X+Y}^2$

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