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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