Correlation between two overlapping flip windows

Question

A fair coin is flipped many times. Let A be the number of heads in flips 1 through 40, and B the number of heads in flips 25 through 64. The two windows each span 40 flips and overlap on flips 25 through 40. What is the correlation between A and B?

Accepted Answer

Split each count into the shared overlap and a private, non-overlapping part. The overlap is flips 25 through 40, which is 16 flips. Write A = C + PA and B = C + PB, where C is the head count in the overlap and PA, PB are the private parts (flips 1 through 24 and flips 41 through 64). The three pieces are mutually independent. With per-flip variance 	frac14: 	ext{Var}(A) = 	ext{Var}(B) = 40 \cdot 	frac14 = 10, and 	ext{Var}(C) = 16 \cdot 	frac14 = 4. Covariance. Only the shared piece contributes: 	ext{Cov}(A, B) = 	ext{Cov}(C + PA, C + PB) = 	ext{Var}(C) = 4. Correlation. ho = \fra…

Correlation between two overlapping flip windows

Your answer

More Probability questions