91Ó°ÊÓ

X and Y are arbitrary random variables for which the necessary expectation and variance exist.

Let \(E\left( Y \right) = \mu Y.\) then

\(\begin{align}Var\left( Y \right) &= E\left( {{Y^2}} \right) - E{\left( Y \right)^2}\\Var\left( Y \right) &= E\left( {{{\left( {Y - \mu Y} \right)}^2}} \right)\\ &= E\left\{ {{{\left( {\left( {Y - E\left( {Y|X} \right)} \right) + \left( {E\left( {Y|X} \right) - \mu Y} \right)} \right)}^2}} \right\}\end{align}\)

\(\begin{align} &= E\left\{ {{{\left( { + \left( {E\left( {Y|X} \right) - \mu Y} \right)} \right)}^2}} \right\}\\ &= E{\left( {Y - E\left( {Y|X} \right)} \right)^2} + 2E\left\{ {\left( {Y - E\left( {Y|X} \right)} \right)\left( {E\left( {Y|X} \right) - \mu Y} \right)} \right\} + \\E\left\{ {{{\left( {E\left( {Y|X} \right) - \mu Y} \right)}^2}} \right\}\end{align}\)

The three expectations in the total sum.

\(\begin{align}E\left\{ {{{\left( {Y - E\left( {Y|X} \right)} \right)}^2}} \right\} &= E\left( {E\left\{ {\left( {Y - E{{\left( {Y|X} \right)}^2}|X} \right)} \right\}} \right)\\ &= E\left( {Var\left( {Y|X} \right)} \right)\end{align}\)

Next,

\(\begin{align}E\left\{ {\left( {Y - E\left( {Y|X} \right)} \right)\left( {E\left( {Y|X} \right) - \mu Y} \right)} \right\}\\ = E\left( {E\left\{ {\left( {Y - E\left( {Y|X} \right)} \right)\left( {E\left( {Y|X} \right) - \mu Y} \right)|X} \right\}} \right)\end{align}\)

\(\begin{align} &= E\left( {\left( {E\left( {Y|X} \right) - \mu Y} \right)E\left\{ {Y - E\left( {Y|X} \right)|X} \right\}} \right)\\ &= E\left( {\left( {E\left( {Y|X} \right) - \mu Y} \right) \times 0} \right)\\ &= 0\end{align}\)

Finally, since the mean \(E\left( {Y|X} \right)\) is \(E\left( {E\left( {Y|X} \right)} \right) = \mu Y\)

\(E\left\{ {{{\left( {E\left( {Y|X} \right) - \mu Y} \right)}^2}} \right\} = {\mathop{\rm var}} \left( {E\left( {Y|X} \right)} \right).\)

It now follows that

\(Var\left( Y \right) = E\left( {Var\left( {Y|X} \right)} \right) + Var\left( {E\left( {Y|X} \right)} \right).\)

Hence proved the theorem that if X and Y are arbitrary random variables for which the necessary expectation and variance exist, then

\(Var\left( Y \right) = E\left( {Var\left( {Y|X} \right)} \right) + Var\left( {E\left( {Y|X} \right)} \right).\)