Compute Gaussian Probability and Regression Coefficients

Suppose $X$ and $Y$ are independent standard normal random variables.

1. Calculate $P[X > 3Y]$.
2. For ordinary linear regression with design matrix $X \in \mathbb{R}^{n \times p}$, response vector $y \in \mathbb{R}^n$, and coefficient vector $\beta \in \mathbb{R}^p$, what is the estimator for $\beta$?
3. When $n$ is much larger than $p$, what is an efficient and exact way to compute $\beta$? Overview: This question tests probabilistic reasoning about Gaussian variables and parameter estimation in ordinary least squares regression, including deriving probabilities for linear combinations of standard normals and computing regression coefficients using linear algebra.