Extending the variance of the sum rule. For mathematical convenience we first extend the sum rule to three random variables with zero expectation. Next we further extend the rule to three random...

Extending the variance of the sum rule. For mathematical convenience we first extend the sum rule to three random variables with zero expectation. Next we further extend the rule to three random variables with nonzero expectation. By the same line of reasoning we extend the rule to n random variables.

a. Let X, Y and Z be random variables with expectation 0. Show that

b. Now show a for X, Y , and Z with nonzero expectation

c. Derive a general variance of the sum rule, i.e., show that if X₁, X₂,...,X_n
are random variables, then