Correlation between overlapping coin-flip counts

Question

A fair coin is flipped 100 times. Let H{50} be the number of heads in the first 50 flips and H{100} the number of heads in all 100 flips. What is the correlation between H{50} and H{100}?

Accepted Answer

Decompose H{100} = H{50} + G, where G (heads in flips 51–100) is independent of H{50}, and both are \mathrm{Bin}(50, 	frac12) with variance 50 \cdot 	frac12 \cdot 	frac12 = 	frac{25}{2}. Covariance. \mathrm{Cov}(H{50}, H{100}) = \mathrm{Cov}(H{50}, H{50}) + \mathrm{Cov}(H{50}, G) = \mathrm{Var}(H{50}) + 0 = \frac{25}{2}. Correlation. ho = \frac{\mathrm{Var}(H{50})}{\sqrt{\mathrm{Var}(H{50})\,\mathrm{Var}(H{100})}} = \sqrt{\frac{\mathrm{Var}(H{50})}{\mathrm{Var}(H{100})}} = \sqrt{\frac{50}{100}} = \frac{1}{\sqrt{2}} \approx 0.707. General rule. For counts over a sub-window of length m in…

Correlation between overlapping coin-flip counts

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