Let $\small X$ be a random variable and it's expectation exist.
$\small Var(X)=E(X-\bar X)^2=E(X^2)-[E(X)]^2$
Important Results:
1. If C is constant then
=$\small V(C)=E(C^2)-[E(C)]^2=C^2-C^2=0$
2.If $\small Y=bX$ then
=$\small V(Y)=V(bX)=b^2E(X^2)-b^2[E(X)]^2=b^2V(X)$
3.If $\small Y=a+bX$ then
=$\small V(Y)=V(a+bX)=E(a^2+2abX+b^2X^2)-[E(a+bX)]^2$
=$\small E(a^2)+2abE(X)+b^2E(X^2)-[a+bE(X)]^2$
=$\small a^2+2abE(X)+b^2E(X^2)-[a^2+2abE(X)+b^2(E(X))^2]$
=$\small b^2V(X)$
4.$\small V(X+Y)=V(X)+V(Y)+2Cov(X,Y)$
Let, $\small Z=X+Y$
$\small V(Z)=E(Z^2)-[E(Z)]^2$
1st Case:$\small E(Z^2)=E(X^2+2XY+Y^2)=E(X^2)+2E(XY)+E(Y^2)$
2nd Case:$\small [E(Z)]^2=E(X)^2+2E(X)E(Y)+E(Y)^2$
$\small V(Z)=E(X^2+2XY+Y^2)=E(X^2)+2E(XY)+E(Y^2)-E(X)^2-2E(X)E(Y)-E(Y)^2$
=$\small V(X)+2Cov(X,Y)+V(Y)$
Alternative:
=$\small E[(X+Y) - (\bar X+\bar Y)]^2$
=$\small E[(X - \bar X)+(Y-\bar Y)]^2$
=$\small E[(X-\bar X)]^2+E[(Y-\bar Y)]^2+2E[(X-\bar X)(Y-\bar Y)]$
=$\small V(X)+V(Y)+2Cov(XY),[E{(X-\bar X)(Y-\bar Y)}=Cov(XY)]$
5.$\small V(X-Y)=V(X)+V(Y)-2Cov(X,Y)$
=$\small E[(X-Y) - (\bar X-\bar Y)]^2$
=$\small E[(X - \bar X)-(Y-\bar Y)]^2$
=$\small E[(X-\bar X)]^2+E[(Y-\bar Y)]^2-2E[(X-\bar X)(Y-\bar Y)]$
=$\small V(X)+V(Y)-2Cov(XY),[E[(X-\bar X)(Y-\bar Y)]=Cov(X,Y)]$
6.$\small V(\sum_{i=1}^{n}X_i)=\sum_{i=1}^{n}V(X_i)+2\sum_{i=1}^{n}\sum_{j=1}^{n}Cov(X_iX_j)$,$\small i<j$
Let, $\small Z=\sum_{i=1}^{n} X_i=X_1+X_2+......+X_n$
$\small V(Z)=E(Z^2)-[E(Z)]^2$
$Z^2= (\sum_{i=1}^{n}X_i)^2=(X_1+X_2+...+X_n)^2$
=$\small (X^2_1+X^2_2+...+X^2_n)+{2(X_1X_2+X_1X_3+...+X_1X_n)+2(X_2X_3+X_2X_4+..+X_2X_n)+...+2(X_{n-1}X_n)}$
=$\small (\sum_{i=1}^{n}X^2_i)+2\sum_{i=1}^{n}\sum_{j=1}^{n}(X_iX_j),i<j$
=$\small E(Z^2)=(\sum_{i=1}^{n}E[X^2_i])+2\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_iX_j),i<j$
$\small E(Z)=E(X_1)+E(X_2)+....+E(X_n)$
or,$\small [E(Z)]^2=[E(X_1)+E(X_2)+....+E(X_n)]^2$
=$\small \sum_{i=1}^{n}[E(X_i)]^2+2\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_i)E(X_j),i<j$
$\small V(Z)=E(Z^2)- [E(Z)]^2$
=$\small (\sum_{i=1}^{n}E[X^2_i])+2\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_iX_j)-\sum_{i=1}^{n}[E(X_i)]^2+2\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_i)E(X_j),i<j$
=$\small \sum_{i=1}^{n} [E{(X^2_i)}-[E(X_i)]^2]+2\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_iX_j)-2\sum_{i=1}^{n}\sum_{j=1}^{n}E(X_i)E(X_j),i<j$
=$\small \sum_{i=1}^{n} [V(X_i)]+2\sum_{i=1}^{n}\sum_{j=1}^{n}Cov(X_iX_j),i<j$
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