Normal marginals imply bivariate normal?

Question

I have a question arising from the answer of this post of mine: Minimal sufficient statistic for simple correlated model That suppose $Y \sim N(\alpha,\beta^2 \tau^2 + \sigma^2)$ ,$X \sim N(0,\tau^2)$ and $\epsilon \sim N(0,\sigma^2)$ where $Y=\alpha+\beta X+\epsilon$ , how can you say that $(X,Y)$ follows bivariate normal distribution? There are examples that the marginals are normal but still the joint distribution is not bivariate normal. We cannot directly say that $(X,Y)$ follows bivariate normal right?

Answer 1

One definition of $(X,Y)$ being bivariate normal is "any linear combination of $X$ and $Y$" is normal. In your case where $Y = \alpha + \beta X + \epsilon$ with $X$ and $\epsilon$ independent normal, we have $aX+bY = (a+b\beta)X + b\epsilon + b\alpha$ which is still normal since it is the sum of independent normal random variables and a constant.