Here is the question:
Suppose $X, Y$ are independent $N(0,1)$ random variables. And take the regression of $Y$ against $X.$ What is the relationship between $R^2$ and sample size approximately?
First I am not sure if "independent" is a typo in the original question. In ordinary least squares (OLS), there is no specific relation between $R^2$ and sample size.
Here R2 is defined as: $$R2 = 1 - \dfrac{RSS}{TSS}.$$ $$RSS = \sum(y_i-\hat{y})^2,\ TSS = \sum(y_i-\bar{y})^2.$$ And in the simple linear regress with interception, R2 is exactly correlation between X and Y.
cor(rnorm(5,), rnorm(5))^2
in R. Now do it for $10$, then $20$, then $50$, then $100$. $\endgroup$