# How to compute a conditional expectation

I want to compute a conditionnal expectation, i know that

$$Z=(Z_1,\ldots,Z_p)'$$ where $$Z_j=\Phi ^{-1}(U_j)$$ with $$Z \sim N(0,R(\theta))$$ and $$R(\theta)$$ the $$p \times p$$ positive definite correlation matrix, $$\Phi ^{-1}(U_j)$$ is the standard normal quantile function and $$U\sim[0,1]$$.

I want to show $$E[Z_i|U_j=u_j]=R_{ij}Z_j$$. I tried by using the correlation between $$Z_i$$ et $$Z_j$$

$$R_{ij}=cor(Z_i,Z_j)=E[Z_iZ_j]-0$$ so

$$R_{ij}=E[Z_iZ_j]=E[\Phi ^{-1}(U_i)\Phi ^{-1}(U_j)]=E[E[\Phi ^{-1}(U_i)\Phi ^{-1}(U_j)|U_j=u_j]]=E[\Phi ^{-1}(U_j)E[\Phi ^{-1}(U_i)|U_j=u_j]] =E[Z_jE[Z_i|U_j=u_j]].$$

But i don't know if it's the right way. Any ideas ?

Thanks

## 1 Answer

The task is to compute $$\mathbb E\left[Z_i|U_j \right].$$ Observe that $$\Phi$$ is invertible hence the $$\sigma$$-algebra generated by $$U_j$$ is the same as the $$\sigma$$-algebra generated by $$Z_j$$. We are thus reduced to compute $$\mathbb E\left[Z_i|Z_j \right].$$ Then decompose $$Z_i$$ as $$Z_i-\frac{R_{i,j}}{R_{j,j}}Z_j+ \frac{R_{i,j}}{R_{j,j}}Z_j$$ to get $$\mathbb E\left[Z_i|Z_j \right]=\mathbb E\left[Z_i- R_{i,j} Z_j|Z_j \right]+\mathbb E\left[ R_{i,j} Z_j|Z_j \right].$$ For the first terms, use and show the independence of $$Z_i- R_{i,j} Z_j$$ and $$Z_j$$; the second term is $$R_{i,j}Z_j$$.