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I am currently trying to understand to get an expression for $p(X,Y) = p(X) p(Y|X)$ where $X \sim \mathcal{N}(x;\mu_1,\sigma^2_1)$ $Y|X \sim \mathcal{N}(y;X,\sigma^2_{2})$. I assume the joint $X,Y$ is a bivariate Gaussian (due to how $X$ and $Y|X$ defined) but what about the mean vector and the covariance matrix of this joint distribution? Is there an easy way to find these parameters from $\mu_1, \sigma_1$ and $\sigma_2$?

I am a bit lost here, any help is appreciated. I have no idea how to move on so I have nothing to show as my own attempt.

Bertrille
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  • The mean vector is $\mathbb E[(X,Y)] = (\mu_1,\mu_1)$. In finding the covariance matrix, you can without loss of generality assume $\mu_1=0$ if that helps – Henry Jan 04 '22 at 14:46
  • You may also find life easier if you consider $Z=Y-X$ and show it is independent of $X$ – Henry Jan 04 '22 at 14:49
  • See https://stats.stackexchange.com/questions/139690/marginal-joint-and-conditional-distributions-of-a-multivariate-normal, https://stats.stackexchange.com/questions/372062/marginal-distribution-of-normal-random-variable-with-a-normal-mean, https://stats.stackexchange.com/questions/386318/deriving-the-joint-probability-density-function-from-a-given-marginal-density-fu – kjetil b halvorsen Jan 04 '22 at 14:50
  • Thank you @Henry!, Thank you @kjetil b halvorsen! Since the link kjetil b halvorsen provides the answer (https://stats.stackexchange.com/questions/372062/marginal-distribution-of-normal-random-variable-with-a-normal-mean) I will close the question. – Bertrille Jan 04 '22 at 15:00
  • According to https://stats.stackexchange.com/a/71303/919, there is a visual, intuitive, and extremely easy way to obtain the answer. – whuber Jan 04 '22 at 15:52

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