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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 $µ$ 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 $U \sim N(0,v^2)$. 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 $Y_2$. But how can I found that? Normally I would integrate by $Y_1$ to find the marginal distribution of $Y_2$ and integrate by $Y_2$ to find the marginal distribution of $Y_1$. But here I dont have an expression with both $Y_1$ and $Y_2$?

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No integrals are needed. Linear combinations of gaussians are still gaussian thus all you have to do is to calculate mean and variance of $Y_2$ and I explained you how to fo that in my answer to your previous question

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  • $\begingroup$ Yes now I have found that $Y_2$ is distributed by $N(\alpha+\beta \cdot \mu,\beta \cdot \sigma^2+v^2)$. So now I have to plot in the new expressions of $\mu$, $\sigma^2$, $\alpha$ and $v^2$? Or what do I have to do? $\endgroup$
    – Lifeni
    Mar 20, 2021 at 15:15
  • $\begingroup$ I have found that on wikipedia that the marginal distribution is given by $p_{X}(x)=E_Y[P_{X|Y}(x|y)]$. And I have earlier found that $E(Y_2|Y_1)=\alpha+\beta \cdot Y_1$. Is that the same and can I use this? $\endgroup$
    – Lifeni
    Mar 20, 2021 at 16:44

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