Heads-Minus-Tails Distribution
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
18-4-2021
2110
Heads-Minus-Tails Distribution
A fair coin is tossed an even 
number of times. Let 
be the absolute difference in the number of heads and tails obtained. Then the probability distribution is given by
![P(D=2k)=<span style=]() {(1/2)^(2n)(2n; n) for k=0; 2(1/2)^(2n)(2n; n+k) for k=1, 2, ..., " src="https://mathworld.wolfram.com/images/equations/Heads-Minus-TailsDistribution/NumberedEquation1.gif" style="height:78px; width:287px" /> |
(1)
|
where 
. The most probable value of 
is 
, and the expectation value is
The generating function for 
is given by
(OEIS A001803 and A046161; Abramowitz and Stegun 1972, Prévost 1933; Hughes 1995). These numbers also arise in one-dimensional random walks.
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."a
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