If we multiply directly, then the new mean is $\mu = (\mu_1 \cdot \mathrm{std}_2^2 + \mu_2 \cdot \mathrm{std}_1^2)/(\mathrm{std}_1^2 + \mathrm{std}_2^2)$ and $\mathrm{std}^2$ is $(\mathrm{std}1^2 + \mathrm{std}_2^2)$.

suppose we have on simple formula, that is "distance = speed $\cdot$ time", where mean of speed is $5$ and mean of time is $3$. Then mean of the distance would be $15$. If we transfer this formula in distribution, with some standard deviation.

But frequently, take $\mathrm{std}_1 = 1$ and $\mathrm{std}_2 = 1$. Then apply mean and standard deviation in the main above formula . The calculated mean and $\mathrm{std}^2$ is $7.5$ and $0.5$. The calculated mean is far away from the distance value.

  • $\begingroup$ Are $X$ and $Y$ independent? If so, I don't see where your formulas come from. $\endgroup$ – kimchi lover Mar 22 at 15:52

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