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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.

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