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Heads-Minus-Tails Distribution

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HeadsMinusTails

A fair coin is tossed an even 2n number of times. Let D=|H-T| be the absolute difference in the number of heads and tails obtained. Then the probability distribution is given by

P(D=2k)={(1/2)^(2n)(2n; n) for k=0; 2(1/2)^(2n)(2n; n+k) for k=1, 2, ...,
(1)

where P(D=2k-1)=0. The most probable value of D is D=2, and the expectation value is

<D_n>=(n(2n; n))/(2^(2n-1))
(2)
=(2Gamma(1/2+n))/(sqrt(pi)Gamma(n))
(3)
=(2sqrt(pi)(-1)^n)/(Gamma(n)Gamma(1/2-n))
(4)
=(-1)^(n+1)(-3/2; -(n+1)).
(5)

The generating function for <D> is given by

sum<D_n>x^n=x/((1-x)^(3/2))
(6)
=1+3/2x+(15)/8x^2+(35)/(16)x^3+...
(7)

(OEIS A001803 and A046161; Abramowitz and Stegun 1972, Prévost 1933; Hughes 1995). These numbers also arise in one-dimensional random walks.

See also

Bernoulli Distribution, Coin, Coin Tossing, Random Walk--1-Dimensional

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 798, 1972.Handelsman, M. B. Solution to Problem 436. "Distributing 'Heads' Minus 'Tails.' " College Math. J. 22, 444-446, 1991.Hughes, B. D. Eq. (7.282) in Random Walks and Random Environments, Vol. 1: Random Walks. New York: Oxford University Press, p. 513, 1995.Prévost, G. Tables de Fonctions Sphériques. Paris: Gauthier-Villars, pp. 156-157, 1933.Sloane, N. J. A. Sequences A001803/M2986 and A046161 in "The On-Line Encyclopedia of Integer Sequences."

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Heads-Minus-Tails Distribution

Cite this as:

Weisstein, Eric W. "Heads-Minus-Tails Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Heads-Minus-TailsDistribution.html

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