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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 $ฯƒ^2$ 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 $๐‘_๐‘‹(๐‘ฅ)=๐ธ_๐‘Œ[๐‘ƒ_{๐‘‹|๐‘Œ}(๐‘ฅ|๐‘ฆ)]$ and I have in a previous task found that $๐ธ(๐‘Œ_2|๐‘Œ_1)=๐›ผ+๐›ฝโ‹…๐‘Œ_1$. 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 $Y_2$?

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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$

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