Let a random variable X be normal $N(\mu,\sigma^2)$ and let the conditional distribution of Y given X be normal $N(a_1+a_2X,\sigma_1^2)$. Find the marginal distribution of Y.
I found their joint density by multiplying the conditional density of Y given X and the density of X. But I couldn't integrate it wrt x to get marginal density of Y. Then I realized that $Y|X-a_1-a_2X$ is $N(0,\sigma_1^2)$, but I couldn't proceed after that.