# Transformation of a normal distribution

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?

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

• Nice, that makes sense. But how did you found $E(Y_1,Y_2)$, when you found the covariance? Mar 19, 2021 at 10:02
• @Lifeni : substitute $Y_2=\alpha+\beta Y_1+U$, expand and find the expectation in terms of $Y_1$. If my answer has been useful you can accept it...I would appreciate Mar 19, 2021 at 11:34
• Thanks you that makes sense :-) Mar 19, 2021 at 11:51