by csy829
Last Updated August 14, 2019 15:20 PM

I was taught that combined variance of two independent variables will be the sum of their variances. However, I wonder if given two sets of independent random variables, say $\{X_m\}$ have mean $A$ and variance $B$, and $\{Y_n\}$ have mean $A$ and variance $C$. If the two groups are mixed together, what will be the new variance?

I'm going to assume you're describing a mixture. That is, suppose you choose the random variable (call it $Z$) from $X$ or $Y$ with a fixed selection probability, where I'll also assume the selection probability is ${1 \over 2}.$

The mean is still $A.$ Calculate the second moment conditionally, knowing that the second moment is the sum of the squared mean and the variance: $$E[Z^2]= {1 \over 2} \left( A^2 + B \right) + {1 \over 2} \left( A^2 + C \right) =A^2 + {1 \over 2} \left(B + C \right) $$

Now find the variance: $$\sigma_Z^2 = E[Z^2]-\mu^2={1 \over 2} \left(B + C \right)$$

The more general formula can be found here http://en.wikipedia.org/wiki/Mixture_distribution in the moments section.

This is for a mixture. Note that the lesson you were taught is for the sum of $X$ and $Y.$

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