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I'm trying to derive the gaussian conditional distribution for a 2 variable gaussian. I'm doing this as I'm studying Gibbs and need to learn how to derive conditionals the analytic way; practice makes perfect.

The joint dist of x1 and x2:

bivariate

The dist of x2:

univariate

The conditional dist: conditional

I'm completely lost on simplifying the equation. I understand that the conditional of a normal is also a normal, but I'm not "seeing it" or even seeing how to move forward. My best guess is simplifying terms in the exponents. Buy in order to do so, I need to make the constant -1/2(1-P^2) go away in the joint. I'm not sure but I believe I can pull this out, as the natural log of this value.

If so, this would allow the terms to consolidate somewhat. Is this the right direction? Am I thinking about it wrong? (I am trying to solve this brute force analytically, not through a short cut which requires a proof of its own to understand.)

Edit: When I do as described above, factor the multiplied term out of the exponent, I end up with:

simplified

Not sure that this is beneficial. However, it does look slightly improved..

jbuddy_13
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  • For examples of the algebra involved, see [our threads on completing the square](https://stats.stackexchange.com/search?q=complete+square). For the duplicate I found one that does not use matrix notation and explains the steps. – whuber Sep 08 '20 at 14:56
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    Keep in mind that your purpose is to show x1 (under a given x2) is a normal distribution, and treat x2 as a constant. So, what you need to do is try to move any term with x2 alone out of the exponents, and leave the terms containing x1. In addition, "make the constant -1/2(1-P^2) go away" is not an issue, since it can be combined with σ_1 as a part of the denominator of the exponent. You should be able to reach there. Good luck. – user295357 Sep 08 '20 at 15:58
  • Re the edit: the log should not appear there. You need to focus on the argument of $\exp,$ which is a quadratic function. All you need to do is identify the coefficients of $x_1^2,$ $x_1,$ and $1$ in the argument. That requires only polynomial algebra, nothing more. An effective approach is literally to read those coefficients off of the original expression, scanning from term to term to look for anything that will form an $x_1^2,$ then again for the multiples of $x_1,$ and finally for everything that is a constant. – whuber Sep 08 '20 at 17:07

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