I am working on a problem where I must show that the conditional distribution of Y given X follows the distribution with mean and variance shown below. In the previous question, we were given that X and Y form a bivariate normal distribution and I derived the marginal distribution for X and Y. I am stuck as to how to proceed with this question.
Show that the conditional distribution of Y given X is: **
$Y|X = x \sim N(\alpha + \beta x, \sigma_{Y|x}^2$, where $\alpha = \mu_Y - \mu_{X}p\frac{\sigma_Y}{\sigma_X}, \beta=p \frac{\sigma_Y}{\sigma_X},$ and $\sigma^2_{Y|X} = \sigma^2_{Y}(1-p^2).$
I know the conditional is $\frac{f(y,x)}{f(x)}$. Is it safe to assume that Y|X form a linear relationship and thus the expectation?
