4.4 Conditional Variance
Definition 4.4.1. Let \(X\) and \(Y\) be random variables with joint probability function. The variance of \(X\) given \(Y\) is denoted by \(V(X/Y)\) is given as \[\Var (X \mid Y)= E(X^2 \mid Y)-\Big (E(X \mid Y)\Big )^2\] \[\Var (X \mid Y)=E\Big \{X/Y-E(X \mid Y)\Big \}^2\]
Result 4.4.2. Let \(X\) and \(Y\) be random variables with joint probability function. Then the variance of \(X\), written \(\Var (X)\), is given by \[\Var (X)=E(\Var (X \mid Y))+\Var (E(X \mid Y)).\]
Proof. \(\Var (X \mid Y) =E(X^2 \mid Y)-\Big (E(X \mid Y)\Big )^2\) \begin {align*} \therefore \hspace {0.4cm} E\Big (\Var (X \mid Y)\Big ) &=E(E(X^2 \mid Y))-E\Big (E(X \mid Y)\Big )^2\\ &=E(X^2)-E\Big (E(X \mid Y)\Big )^2............. (1) \end {align*}
\begin {align*} \Var (E(X \mid Y)) &=E\Big (\Big (E(X \mid Y)\Big )^2\Big )-\Big (E(E(X \mid Y)\Big )^2\\ &=E\Big (\Big (E(X \mid Y)\Big )^2\Big )-\Big (E(X)\Big )\\ \end {align*}
\[E\Big (\Big (E(X \mid Y)\Big )\Big )^2=\Var (E(X \mid Y))+\Big (E(X)\Big )^2................(2)\]
substituting in (1) we get \(E(\Var (X \mid Y))=E(X^2)-\Var (E(X \mid Y))-\Big (E(X)\Big )^2\)
\[E(\Var (X \mid Y))+\Var (E(X \mid Y))=E(X^2)-\Big (E(X)\Big )^2=\Var (X).\]
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Example 4.4.3. Suppose that by time \(t\) the number of passengers that have checked in for a
particular flight is a Poisson random variable with mean \(\mu t\). If the checking in counter closes at
time independent of the passengers arriving at the counter that is uniformly distributed over \((0,T)\).
Find the mean and variance of the number of passengers who would have checked in by time \(t\).
Solution. Let \(X\) be the random variable for the time the counter closes and \(N(X)\) be the random variable of the number of passengers who would have checked in if the counter closed at time \(X\). \begin {align*} E(N(X)/X=t) &=\mu t.\\\\ E(N(X)/X=t)&=E(N(t)/X=t)=E(N(t)) \end {align*}
\begin {align*} E(N(X)) &=E(E(N(X)/X=t))\\ &=E(\mu t)\\ &=\mu E(t)\\ &=\mu \frac {T}{2},\hspace {0.4cm}\text {Thus the mean}\\\\ \end {align*}
\[\Var (N(X)/X=t)=\mu t\] \begin {align*} \implies \hspace {0.5cm} \Var (N(X)) &=E(\Var (N(x)/X=t))+\Var (E(N(X)/X=t))\\ &=E(\mu t)+\Var (\mu t)\\ &=\frac {\mu T}{2}+\mu ^2 \Var (t)\\ &=\frac {\mu T}{2}+\mu ^2\Var (X)\\ &=\frac {\mu T}{2}+\frac {\mu ^2 T^2}{12}\\ &=\frac {\mu T}{12}(6+\mu T)\hspace {0.5cm} \text {variance}\\\\\\ \end {align*}
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