Let a random variable $X$ be normal $N(\mu,\sigma^2)$ and let the conditional distribution of $Y$ given $X$ be normal $N(a_1+a_2X,\sigma_1^2)$.
a)Find the joint probability density function of $X$ and $Y$.
b)Find the marginal distribution of $Y$ and the correlation coefficient of $X$ and $Y$.
For (a), I just multiplied the conditional density of $Y$ given $X$ and density of $X$; and I think it's ok. For (b), I tried to write their joint density in the form of bivariate normal but couldn't do that. On the other hand, we know that if the random variables $X$ and $Y$ are bivariate normal then, the conditional distribution of $X$ given $Y$ is normal with mean $E[X\mid Y]$ and variance $(1-\rho^2)\sigma_X^2$. But is that true that if $X$ is normal and $Y$ given $X$ is normal, then they are bivariate normal? So that I can do (b) easily, or is there another way to solve this question?