Linear model what is $p(x|y_0)$

Question

If I have a linear model of the form:
$$x_i = beta y_i + alpha + epsilon_i$$
where $epsilon_i$ are samples from $epsilon$, an independent and identically distributed random variable. I can find estimates $hat{beta}, hat{alpha}$ using ordinary least squares.
Given a value of $y=y_0$, can I write
$$p(x|y_0) = hat{beta} y_0 + hat{alpha} + p(epsilon)$$

chasmani · Answer

Answering my own question:

No, you cannot write that. You can however consider $x|y_0$ as a translation of the random variable $epsilon$. So

$$x|y_0 = hat{beta} y_0 + hat{alpha} + epsilon$$

In expectation:

$$<x|y_0> = hat{beta} y_0 + hat{alpha} + <epsilon>$$

And the higher moments will be the same as the moments of the noise distribution.

So $p(x|y_0)$ will have the same shape as $p(epsilon)$, but shifted along the axis by $hat{beta} y_0 + hat{alpha}$.

Linear model what is $p(x|y_0)$

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