# Understanding jointly Gaussian distribution [duplicate]

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.

• 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 Commented Jan 4, 2022 at 14:46
• You may also find life easier if you consider $Z=Y-X$ and show it is independent of $X$ Commented Jan 4, 2022 at 14:49
• Commented Jan 4, 2022 at 14:50
• Thank you @Henry!, Thank you @kjetil b halvorsen! Since the link kjetil b halvorsen provides the answer (stats.stackexchange.com/questions/372062/…) I will close the question. Commented Jan 4, 2022 at 15:00
• According to stats.stackexchange.com/a/71303/919, there is a visual, intuitive, and extremely easy way to obtain the answer.
– whuber
Commented Jan 4, 2022 at 15:52