Finding the marginal distributions

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

The hourly wage for period 1 is normally distributed with mean $$ยต$$ and variance $$ฯ^2$$ 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 $$U \sim N(0,v^2)$$. Then we assume that $$\beta \neq 0$$ and let $$\mu = 350$$ and $$\sigma^2=12365$$ and $$\alpha=350\cdot(1-\beta)$$ and $$v^2=12365 \cdot (1-\beta^2)$$.

Now I have to find the marginal distributions of $$Y_1$$ and $$Y_2$$. I have found on Wikipedia that the marginal probability is $$๐_๐(๐ฅ)=๐ธ_๐[๐_{๐|๐}(๐ฅ|๐ฆ)]$$ and I have in a previous task found that $$๐ธ(๐_2|๐_1)=๐ผ+๐ฝโ ๐_1$$. Is that the same value (I'm not totally sure on the notation will be the same) and how can I use it to find the marginal distributions of $$Y_1$$ and $$Y_2$$?

in your problem the issue is simplier. You just have a stated linear relation that links $$Y_1$$ and $$Y_2$$. On the internet you can also find that "linear combination of Gaussians are still Gaussian" and your $$Y_2$$ is a linear function of $$Y_1$$ and $$U$$ thus it is also gaussian. All you have to do is to substitute values and use the properties of Expectation and Variance

• $$\mathbb{E}[a+bX]=a+b\mathbb{E}[X]$$ (Expectation is a linear operator)

• $$\mathbb{V}[a+bX+Y]=b^2\mathbb{V}[X]+\mathbb{V}[Y]$$ (if X and Y are incorrelated)

Thus

$$\mathbb{E}[Y_2]=350-350\beta+350\beta+0=350$$

$$\mathbb{V}[Y_2]=\beta^2 12365+12365(1-\beta^2)=12365$$

concluding, marginal distribution are

$$Y_1\sim N(350;12365)$$

$$Y_2\sim N(350;12365)$$

...they are identically distributed

If you do a step forward you can see that they are identically distributed but correlated with correlation coefficient $$\rho=\beta$$