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I am trying to understand the following statement:

enter image description here

Can someone please explain what is meant by "the conditional expectation function m(x) is linear in x"?

In the case of regression, I understand the idea behind "linearity" - a linear function can be created using the model parameters and covariates, that is linked to the response in a "linear" way : E(y) = g(X-transposeBeta) ... where the function "g" can be considered as a "linear function". Thus, we can say that y = b0 + b1x1 + b2x2 +....bnxn , i.e. "linear".

But in the above statement, if you have a joint multivariate normal distribution between a response variable "Y" and covariates "X1, X2 ... Xn" : P(Y, X1, X2...Xn) ~ MVN

From the above equation, if you wanted to find out the conditional expectation : E(Y | X1 = x1, X2 = x2, ... Xn = xn) : How do we know that E(Y | X1 = x1, X2 = x2, ... Xn = xn) is linear in x?

Can someone please explain this? What does it even mean to be "linear" in this case?

Thanks!

stats_noob
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    In the MVN case, it follows from the definition of conditional expectation applied the MVN. Try it for the bivariate case first. – Glen_b Dec 06 '21 at 03:59
  • @ Glen_b: Thank you for your reply! Can you please explain a bit more? What exactly is meant by "linear" here? Thanks! – stats_noob Dec 06 '21 at 05:21
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    When $(Y,\boldsymbol X)$ is jointly normal, $E[Y\mid \boldsymbol X]=\alpha+ \boldsymbol \beta^T\boldsymbol X$ for some suitable scalar $\alpha$ and vector $\boldsymbol \beta$. See https://stats.stackexchange.com/a/30600/119261. – StubbornAtom Dec 06 '21 at 11:34

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