Let X and Y have a bivariate normal distribution with parameters $\mu_X=-3$,$\mu_Y=10$, $\sigma^2_X=25$, $\sigma^2_Y=9$, and $\rho=\frac{3}{5}$. Find the distribution of X and conditional distribution of X, given Y=13.
Question: What's the general strategy for solving distribution and conditional distribution?
The [marginal] distribution of $X$ is simply $N(\mu_X, \sigma_X^2)$.
For a bivariate distribution $(X,Y)$ with parameters $\mu_X, \mu_Y, \sigma^2_X, \sigma^2_Y, \rho$, one can equivalently view $X$ as $$\frac{X-\mu_X}{\sigma_X} = \rho \cdot \frac{Y-\mu_Y}{\sigma_Y} + \sqrt{1-\rho^2} \cdot Z$$ where $Z\sim N(0,1)$ is independent of $Y$. You can check that $E[X] = \mu_X$, $\text{Var}(X) = \sigma_X^2$, and $\text{Corr}(X,Y) = \rho$.
With this formulation of $X$, the conditional distribution of $X$ given $Y=y$ is straightforward, since conditioning yields $$\frac{X-\mu_X}{\sigma_X} = \rho \frac{y-\mu_Y}{\sigma_Y} + \sqrt{1-\rho^2} Z \sim N(\rho(y-\mu_Y)/\sigma_Y, 1-\rho^2)$$ $$X \sim N(\mu_X + \sigma_X \rho (y-\mu_Y)/\sigma_Y, \sigma_X^2(1-\rho^2))$$