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I have two random variables $X\sim \mathcal{N}(\bar{x},\sigma_x)$ and $Y\sim \mathcal{N}(\bar{y},\sigma_y)$. I denote $\alpha_1$ the linear regression coefficient of $Y$ versus $X$, and $\alpha_2$ the linear regression coefficient of $X$ versus $Y$.

How do I find the distribution of the product $\alpha_1 \times \alpha_2$ ?

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Suppose we let $s_X$ and $s_Y$ denote the sample standard deviation of the two variables and we let $r_{XY}$ denote the sample correlation. For simple linear regression with an intercept term, the coefficient estimators are:

$$\hat{\alpha}_1 = r_{XY} \cdot \frac{s_Y}{s_X} \quad \quad \quad \quad \quad \hat{\alpha}_2 = r_{XY} \cdot \frac{s_X}{s_Y},$$

so their product is:

$$\hat{\alpha}_1 \times \hat{\alpha}_2 = r_{XY}^2 = R^2.$$

That is, the product of the regression coefficients from these two inverse-models is the square of their sample correlation, which is the coefficient-of-determination for either regression model. As to its distribution, that depends on the joint distribution of the values in your model, which you haven't specified. It is possible to derive the distribution of this quantity under various assumptions about the underlying distribution of the values going into the regression (see e.g., here).

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