I also post this question in CrossValidate. Consider the following problem:
Let $X, Y$ be two independent uniform random variable on $(0,1)$. We consider the regression model $Y = \beta_1 X + \beta_0$, given the restriction that $X + Y > 1$. Find $\beta_1$.
Let us call $U = X|X + Y > 1$, $V = Y|X+Y > 1$. Then we could compute the density of $U$ and $V$ by doing conditional probability, and $\beta_1 = \frac{Cov(U,V)}{Var(U)}$, $\beta_0$ could be found by plugging in the mean of $U$ and $V$. The final answer should be $\beta_1 = -\frac{1}{2}$.
My question is: how to compute $\beta_1$ based on geometry? If we draw the graph, we could see that we are doing a right-top triangle within the unit square $[0,1] \times [0,1]$. We could also know the center of the triangle is $(\frac{2}{3}, \frac{2}{3}) = (E[U], E[V])$. Is there a way that we could visualize the $\beta_1$ should be $-\frac{1}{2}$?