91Ó°ÊÓ

Show that$Cov(X,Yâˆ\pounds Z)≡E\left[\right(X-E[Xâˆ\pounds Z]\left)\right(Y-E[Yâˆ\pounds Z])âˆ\pounds Z]$

We are given that,

$\begin{array}{r}\mathrm{Cov}(X,Yâˆ\pounds Z)=E[X−E(Xâˆ\pounds Z\left)\right(Y−E(Yâˆ\pounds Z)âˆ\pounds Z\left)\right]\end{array}$

$\begin{array}{r}⇒\mathrm{Cov}(X,Yâˆ\pounds Z)=E[XY−XE(Yâˆ\pounds Z)−YE(Xâˆ\pounds Z)+E(Xâˆ\pounds Z)â‹\dots E(Yâˆ\pounds Z)âˆ\pounds Z]\end{array}$

$=E(XYâˆ\pounds Z)−E\left[XE\right(Yâˆ\pounds Z)âˆ\pounds Z]−E\left[YE\right(Xâˆ\pounds Z)âˆ\pounds Z]+E\left[E\right(Xâˆ\pounds Z\left)E\right(Yâˆ\pounds Z)âˆ\pounds Z]$

$=E[XYâˆ\pounds Z]-E(Xâˆ\pounds Z)·E(Yâˆ\pounds Z)-E(Yâˆ\pounds Z)·E(Xâˆ\pounds Z)+E[Xâˆ\pounds Z]·E[Yâˆ\pounds Z]$

$=E[XYâˆ\pounds Z]-E[Xâˆ\pounds Z]·E[Yâˆ\pounds Z]$

It has been shown that$\mathrm{Cov}(X,Yâˆ\pounds Z)=E[XYâˆ\pounds Z]−E[Xâˆ\pounds Z]E[Yâˆ\pounds Z].$

The conditional covariance formula$=\mathrm{Cov}(X,Y)=E\left[\mathrm{Cov}\right(X,Yâˆ\pounds Z\left)\right]+\mathrm{Cov}\left(E\right[Xâˆ\pounds Z],E[Yâˆ\pounds Z\left]\right)$

b) $\text{R.H.S.}=E\left[\mathrm{Cov}\right(X,Yâˆ\pounds Z\left)\right]+\mathrm{Cov}\left[E\right(Xâˆ\pounds Z),E(Yâˆ\pounds Z\left)\right]$

Using Result of part (a)

$\begin{array}{r}=E\left[E\right(XYâˆ\pounds Z)−E(Xâˆ\pounds Z\left)E\right(Yâˆ\pounds Z\left)\right]+E\left[E\right(Xâˆ\pounds Z)â‹\dots E(Yâˆ\pounds Z)âˆ\pounds Z]−E\left[E\right(Xâˆ\pounds Z)âˆ\pounds Z]â‹\dots E\left[E\right(Yâˆ\pounds Z)âˆ\pounds \end{array}$

$\begin{array}{r}=E\left(XY\right)−E\left(X\right)E\left(Y\right)+E\left(X\right)â‹\dots E\left(Y\right)−E\left(X\right)â‹\dots E\left(Y\right)\end{array}$

role="math" localid="1647526707691" $=E\left(XY\right)−E\left(X\right)â‹\dots E\left(Y\right)$

$=\mathrm{Cov}(X,Y)$

Therefore, the conditional covariance formula $\mathrm{Cov}(X,Y)=E\left[\mathrm{Cov}\right(X,Yâˆ\pounds Z\left)\right]+\mathrm{Cov}\left(E\right[Xâˆ\pounds Z],E[Yâˆ\pounds Z\left]\right)$has been proved.

Set$X=Y$in part (b)

c) Putting $X=Y$in result of part (b)

role="math" $\mathrm{Cov}(X,X)=E\left[\mathrm{Cov}\right(X,X)âˆ\pounds Z]+\mathrm{Cov}\left[E\right(Xâˆ\pounds Z),E(Xâˆ\pounds Z\left)\right]$

$⇒\mathrm{Var}\left(X\right)=E\left[\mathrm{Var}\right(X)âˆ\pounds Z]+\mathrm{Var}\left[E\right(Xâˆ\pounds Z\left)\right]$

Similarly$\mathrm{Var}\left(\mathrm{Y}\right)=\mathrm{E}\left[\mathrm{Var}\right(\mathrm{Y})âˆ\pounds \mathrm{Z}]+\mathrm{Var}\left[\mathrm{E}\right(\mathrm{Y}âˆ\pounds \mathrm{Z}\left)\right]$

Therefore, the conditional variance formula is$\mathrm{Var}\left(Y\right)=E\left[\mathrm{Var}\right(Y)âˆ\pounds Z]+\mathrm{Var}\left[E\right(Yâˆ\pounds Z\left)\right]$.