Let $X$ and $Y$ be two continuous random variables. Suppose that:
- $X$ has normal pdf with mean mu_1 and variance $\sigma_1^2$
- $Y|X$ has normal pdf with mean $mu_2=(a+b*X)$ and variance $\sigma_2^2$; where a and b are two constants from linear regression
- $f(Y)=\int f(Y|X)f(X) dx$; where $f()$ is the pdf
Question: How do I show that $f(Y)$ is a normal distribution? How do I determine its parameters based on the information I'm given?
Note: I have tried to brute force multiply the densities and integrate, which yields the answer to my question. However, is there a faster way to see this because I'm interested in extending to the case where $X$ is a random vector?