Let $A$ and $B$ be non-empty disjoint sets of real numbers. Suppose that the average of the numbers in the first set is $\mu_{A}$ and the average of the numbers in the second set is $\mu_{B}$; let the corresponding variances be $v_{A}$ and $v_{B}$ respectively. If the average of the elements in $A \cup B$ is $\mu= p.\mu_{A} + (1 - p).\mu_{B}$, what is the variance of the elements in $A \cup B$?
- $p.v_{A}+ (1 - p).v_{B}$
- $(1 - p). v_{A}+ p. v_{B}$
- $p.[v_{A}+(\mu_{A}-\mu)^{2}]+(1 - p). [v_{B}+ (\mu_{B}-\mu)^{2}]$
- $(1 - p).[v_{A}+(\mu_{A}-\mu)^{2}]+ p. [v_{B}+ (\mu_{B}-\mu)^{2}]$
- $p.v_{A}+ (1 - p). v_{B} + (\mu_{A}- \mu_{B})^{2}$