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Rules for Variance

The variance of a constant is zero
Adding a constant value, c to a variable does not change variance (because the expectation increases by the same amount).
- $$ \sigma_{x+c} = VAR(X+c) = E[((X_{i} + c)-E(\overline{X} + c))^{2}] = VAR(X) $$
Multiplying a constant value, c to a variable increase the variance by square of the constant, c.
- $$ \sigma_{c*x} = VAR(cX) = c^{2}VAR(X)$$
The variance of the sum of two or more random variables is equal to the sum of each of their variances only when the random variables are independent.
- $$ VAR(X+Y) = VAR(X) + 2 COV(X,Y) + VAR(Y)$$