Transformation of a normal distribution

Question

I got these 2 models for hourly wage for 2 periods:

The hourly wage for period 1 is normally distributed with mean $µ$ and variance $σ^2$ so $Y_1 \sim N(\mu,\sigma^2)$.

And the hourly wage for period 2 is given by: $$Y_2=\alpha+\beta Y_1+U$$ where $Y_1$ and $U$ are independent and $U \sim N(0,v^2)$

I have to find the distribution for $Y_2$.

So I think if $Y_1$ is normal distributed and U is normal distributed then we get that $Y_2 \sim N(E(Y_2),var(Y_2))$. Have I understood that correct?

Now I have to check if $Y_1$ and $Y_2$ are independent. When $Y_1$ is included in the model for $Y_1$ I think that they are not independent, but what will the formally argument be?

Answer 1

Yes correct.

$$Y_2\sim N(\alpha+\beta\mu;\beta^2\sigma^2+v^2)$$

To check independence you can check covariance because in a Gaussian model, Incorrelation and independence are equivalent

$$\mathbb{Cov}[Y_1,Y_2]=\mathbb{E}[Y_1Y_2]-\mathbb{E}[Y_1]\cdot \mathbb{E}[Y_2]=\beta\sigma^2$$

thus they are not independent