# What is the distribution of the product of regression coefficients?

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 $$\alpha_1 \times \alpha_2$$ ?

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:
$$\hat{\alpha}_1 = r_{XY} \cdot \frac{s_Y}{s_X} \quad \quad \quad \quad \quad \hat{\alpha}_2 = r_{XY} \cdot \frac{s_X}{s_Y},$$
$$\hat{\alpha}_1 \times \hat{\alpha}_2 = r_{XY}^2 = R^2.$$