Part CCSIR NET December 2024the-independence-variance-is-one-over-n-minus-one

The independence variance is one over n minus one

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{n}, Y_{n})$ be a random sample of size n from a bivariate distribution F(X,Y) with absolutely continuous marginal distribution functions F_X and F_Y of X and Y, respectively. Let r_S be the Spearman's rank correlation coefficient defined with ranks of $X_{i}$ and ranks of $Y_{i}, i = 1, 2, \dots, n$ . Then, which of the following statements are true?

A.If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then E(r_S) = 0, ∀n ≥ 2.✓
B.If n = 3, then P(r_S = 0) = 0.✓
C.If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then Var(r_S) = 1/n, ∀n ≥ 2.
D.If n = 4 and F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then P(r_S = 0) = 1/24.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

The independence variance is one over n minus one

More on this topic