Question

Question: Prove the general case of Theorem 7. That is, show that if \({X_1},{X_2}, \ldots ,{X_n}\) are pairwise independent random variables on a sample space \(S\), where \(n\) is a positive integer, then \(V\left( {{X_1} + {X_2} + \cdots + {X_n}} \right) = V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \cdots + V\left( {{X_n}} \right)\). (Hint: Generalize the proof given in Theorem 7 for two random variables. Note that a proof using mathematical induction does not work; sec.

Short answer

Answer

Proved using mathematical induction \(V\left( {{X_1} + {X_2} + \ldots \ldots .. + {X_n}} \right) = V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \ldots .. + V\left( {{X_n}} \right)\)

Step-by-step solution

In the problem given  

\({X_1} + {X_2} + \ldots \ldots .. + {X_n}\) are pairwise independent random variables on a sample space \(s\), where \(n\) is a positive integer.

The definition and the formula for the given problem

Let

\(V\left( {{X_1} + {X_2} + {X_3} + \ldots \ldots \ldots .. + {X_k} + {X_{k + 1}}} \right) = V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \ldots \ldots .. + V\left( {{X_k}} \right) + V\left( {{X_{k + 1}}} \right)\)

It would be assumed that\({X_1} + {X_2} + {X_3} + \ldots \ldots \ldots \ldots . + {X_k}\)and\({x_{k + 1}}\)are independent, which is not true for this case\(\sin \theta ,{X_1} + {X_2}\)and\({X_3}\)are not independent.

Therefore, it would not be possible to proof by mathematical induction.

Determining the sum in expanded form

Variance \(V(X) = E{(X)^2} - {(E(X))^2}\)

\(\begin{aligned}{c} &= \left\{ {E{{\left( {{X_1} + {X_2} + \ldots \ldots \ldots .. + {X_n}} \right)}^2}} \right\} - {\left\{ {E\left( {{X_1} + {X_2} + \ldots \ldots .. + {X_n}} \right)} \right\}^2}\\= E\left( {\begin{aligned}{{}{}}{X_1^2 + X_2^2 + \ldots \ldots \ldots \ldots .... + X_n^2 + {X_1}{X_2} + {X_1}{X_3} + \ldots \ldots \ldots \ldots ... + {X_1}{X_n} + {X_2}{X_1}}\\{ + {X_2}{X_3} + \ldots \ldots \ldots \ldots ... + {X_2}{X_n} + \ldots .. + {X_n}{X_1} + {X_n}{X_2} + \ldots \ldots \ldots ... + {X_n}{X_{n - 1}}}\end{aligned}} \right)\\ - {\left\{ {E\left( {{X_1}} \right) + E\left( {{X_2}} \right) + \ldots \ldots \ldots ... + E\left( {{X_n}} \right)} \right\}^2}\\ &= E\left( {X_1^2} \right) + E\left( {X_2^2} \right) + \ldots \ldots E\left( {X_n^2} \right) + E\left( {{X_1}{X_2}} \right) + E\left( {{X_1}{X_3}} \right) + \ldots ..\\ + E\left( {{X_1}{X_n}} \right) + E\left( {{X_2}{X_1}} \right) + E\left( {{X_2}{X_3}} \right) + \ldots .. + E\left( {{X_2}{X_n}} \right) + \ldots \\ + E\left( {{X_n}{X_1}} \right) + E\left( {{X_n}{X_2}} \right) + \ldots \ldots . + E\left( {{X_n}{X_{n - 1}}} \right) - E{\left( {{X_1}} \right)^2} - E{\left( {{X_2}} \right)^2}\\ \ldots \ldots E{\left( {{X_n}} \right)^2} - E\left( {{X_1}} \right)E\left( {{X_2}} \right) - E\left( {{X_1}} \right)E\left( {{X_3}} \right) \ldots \ldots ..E\left( {{X_1}} \right)E\left( {{X_n}} \right)\\ - E\left( {{X_2}} \right)E\left( {{X_1}} \right) - E\left( {{X_2}} \right)E\left( {{X_3}} \right) \ldots \ldots E\left( {{X_2}} \right)E\left( {{X_n}} \right) \ldots ..\\E\left( {{X_n}} \right)E\left( {{X_1}} \right) + E\left( {{X_n}} \right)E\left( {{X_2}} \right) + \ldots .. + E\left( {{X_n}} \right)E\left( {{X_{n - 1}}} \right)\end{aligned}\)

As \(X\) and \({\rm{Y}}\) are independent, we have \(E(XY) = E(X).E(Y)\)

So,

\(\begin{aligned}{c}V\left( {{X_1} + {X_2} + \ldots \ldots .. + {X_n}} \right) &= E\left( {X_1^2} \right) + E\left( {X_2^2} \right) + \ldots \ldots .. + E\left( {X_n^2} \right) - E\left( {X_1^2} \right) - E\left( {X_2^2} \right) \ldots \ldots .E{\left( {{X_n}} \right)^2}\\\left.{= \left\{ {E\left( {X_1^2} \right) - {{\left( {E\left( {{X_1}} \right)} \right)}^2}} \right\} + \left\{ {E\left( {X_2^2} \right) - {{\left( {E\left( {{X_2}} \right)} \right)}^2}} \right\} + \ldots \ldots . + E\left( {X_n^2} \right) - E{{\left( {{X_n}} \right)}^2}} \right\}\\ &= V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \ldots .. + V\left( {{X_n}} \right)\end{aligned}\)

\(V\left( {{X_1} + {X_2} + \ldots \ldots .. + {X_n}} \right) = V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \ldots .. + V\left( {{X_n}} \right)\)