Correlation of heads in the first quarter versus the whole

Question

A fair coin is flipped 120 times. Let H{30} be the number of heads in the first 30 flips and H{120} the number of heads in all 120 flips. What is the correlation between H{30} and H{120}?

Accepted Answer

Decompose H{120} = H{30} + G, where G (heads in flips 31 through 120) is independent of H{30}. The variances are 	ext{Var}(H{30}) = 30 \cdot 	frac12 \cdot 	frac12 = 7.5 and 	ext{Var}(H{120}) = 120 \cdot 	frac14 = 30. Covariance. Since G is independent of H{30}, 	ext{Cov}(H{30}, H{120}) = 	ext{Cov}(H{30}, H{30}) + 	ext{Cov}(H{30}, G) = 	ext{Var}(H{30}) = 7.5. Correlation. ho = \frac{	ext{Var}(H{30})}{\sqrt{	ext{Var}(H{30})\,	ext{Var}(H{120})}} = \sqrt{\frac{	ext{Var}(H{30})}{	ext{Var}(H{120})}} = \sqrt{\frac{30}{120}} = \sqrt{	frac14} = 	frac12. The sub-window carries a quart…

Correlation of heads in the first quarter versus the whole

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