Intro to Statistical Learning (2nd Ed), Solution to Problem 3.12A (R and Python)

Question 3.12A
This problem involves simple linear regression without an intercept.

(a) Recall that the coefficient estimate β^ for the linear regression of Y onto X without an intercept is given by (3.38).

β^=∑ni=1xiyi∑ni=1x2i

Under what circumstance is the coefficient estimate for the regression of X onto Y the same as the coefficient estimate for the regression of Y onto X?

(b) Generate an example in R/Python with n = 100 observations in which the coefficient estimate for the regression of X onto Y is different from the coefficient estimate for the regression of Y onto X.

(c) Generate an example in R/Python with n = 100 observations in which the coefficient estimate for the regression of X onto Y is the same as the coefficient estimate for the regression of Y onto X.