Finding joint likelihood function for linear regression

Question

Let $Y_i=\alpha_0+\beta_0 X_i + \epsilon_0$, where $\epsilon_i \sim N(0, \sigma_0^2)$ and $X_i \sim N(\mu_x,\tau_0^2)$ are independent.

The data $(X_i, Y_i)$ are generated from $Y_i=\alpha_0+\beta_0 X_i + \epsilon_0$.

I have to find the joint likelihood function, which is given by:

$L_n(${$X_i, Y_i$}$, \alpha, \beta, \mu_x, \sigma^2, \tau^2) = \prod_{i=1}^n f(X_i, Y_i)=\prod_{i=1}^n f_x(X_i)f_{.|X_i}(Y_i)$.

I thought that $f(X_i, Y_i)=\frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \exp \left\{-\frac{1}{2 (1-\rho^2)}\bigg[\bigg(\frac{x-\mu_X}{\sigma_X}\bigg)^2 +\bigg(\frac{y-\mu_Y}{\sigma_Y}\bigg)^2-2\rho \frac{(x-\mu_X)(y-\mu_Y)}{\sigma_X \sigma_Y} \bigg] \right\}$.

But I don't know how to put this in terms of the parameters given or if this is correct. I know that $Y|X \sim (\alpha_0+\beta_0 X, \sigma^2)$. Should I use this for $\mu_Y$ and $\sigma_Y$ such that I have $\alpha$ and $\beta$ in the function?

Thank you in advanced.

Answer 1

Just take the expression you already wrote for the likelihood and plug in the two densities. \begin{align} f_X(X_i) &= \frac{1}{\sqrt{2 \pi \tau_0^2}} e^{-(X_i - \mu_x)^2/2\tau_0^2} \\ f_{\cdot \mid X_i}(Y_i) &= \frac{1}{\sqrt{2 \pi \sigma_0^2}} e^{-(Y_i - (\alpha_0 + \beta_0 X_i))^2/2\sigma_0^2} \end{align}