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Let $\ \mu = \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \Sigma = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $

and I define $\ X_1 = \sigma_1\cdot Z_1 + \mu_1 , \ \ X_2 = \sigma_2(\rho Z_1 + \sqrt{1-\rho^2} Z_2) + \mu_2 $

where $\ Z_1, Z_2 \sim^{iid} N(0,1) $

and I want to find the conditional expectation $\ E[X_2 | X_1 = x_1 ] $

I think in this case it is easy because $\ X_1 = Z_2 , X_2 = Z_2 $ ? but if the covariance matrix isn't the identity matrix and mu isn't $\ 0 $ ?

bm1125
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1 Answers1

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$X_1$ and $X_2$ are jointly normal. By a well known result $E(X_1|X_1)=cX_1$ for some constant $c$. To find the vaue of $c$ use the fact that $E(X_1X_2|X_1)=X_1(E(X_2|X_1))=cX_1^{2}$ so $EX_1X_2=cEX_1^{2}$.

Kavi Rama Murthy
  • 359,332