I am trying to solve the following problem:
Suppose that $\mu\sim N(1,4)$ and $Y|\mu\sim N(\mu,1)$. Show that:
$$\begin{bmatrix}Y \\ \mu \end{bmatrix} \sim N\bigg(\begin{bmatrix}1 \\ 1 \end{bmatrix},\begin{bmatrix}5 & 4 \\ 4 & 4 \end{bmatrix}\bigg) .$$
First, I used the fact that: $$f_{Y,\mu}(Y,\mu)=f_{Y|\mu}(Y|\mu)f_\mu(\mu).$$
Then, I can easily write out $f_\mu(\mu)$ as: $$f_\mu(\mu)=\frac{1}{\sqrt{2\pi}\sqrt{4}}\exp\bigg\{-\frac{(\mu-1)^2}{8}\bigg\}.$$
Now what I do not know is how to write out the conditional density $f_{Y|\mu}(Y|\mu)$ appropriately. I tried to write it as:
$$f_{Y|\mu}(Y|\mu)=\frac{1}{\sqrt{2\pi}}\exp\bigg\{-\frac{(Y-\mu)^2}{2}\bigg\},$$
but, by this, I could not find the right result.
Does any of you have an idea on what is the right way to write out a conditional density? Are my other statements correct?
Thank you in advance for your time!
