“Run”的概念、定义、习题解题方法及翻译-数学辞典

单词

Run

释义

Run

A run is a sequence of more than one consecutive identical outcomes, also known as a Clump. Given Bernoulli Trials (say, in theform of Coin Tossings), the probability of a run of consecutive heads or tails is given by the Recurrence Relation

(1)

where

for

and

(Bloom 1996).

Let denote the number of sequences of indistinguishable objects of type and indistinguishableobjects of type in which no -run occurs. The probability that a -run does occur is then givenby

(2)

where

is a Binomial Coefficient. Bloom (1996) gives the following recurrence sequence for

(3)

where

(4)

Another recurrence which has only a fixed number of terms is given by


	(5)

where

(6)

(Goulden and Jackson 1983, Bloom 1996). These formulas disprove the assertion of Gardner (1982) that``there will almost always be a clump of six or seven Cards of the same color'' in a normal deck of cards by giving

Given Bernoulli Trials with a probability of success (heads) , the expected number of tails is , so the expected number of tail runs is . Continuing,

(7)

is the expected number of runs

. The longest expected run is therefore given by

(8)

(Gordon et al. 1986, Schilling 1990). Given

0s and

1s, the number of possible arrangements with

runs is

(9)

for

an Integer, where

is a Binomial Coefficient. Then

(10)

Bloom (1996) gives the expected number of noncontiguous -runs in a sequence of 0sand 1s as

(11)

where

is the Pochhammer Symbol. For

has an approximately Normal Distribution withMean and Variance

			(12)
			(13)

See also Coin Tossing, Eulerian Number, Permutation, s-Run

References

Bloom, D. M. ``Probabilities of Clumps in a Binary Sequence (and How to Evaluate Them Without Knowing a Lot).'' Math. Mag. 69, 366-372, 1996.

Gardner, M. Aha! Gotcha: Paradoxes to Puzzle and Delight. New York: W. H. Freeman, p. 124, 1982.

Godbole, A. P. ``On Hypergeometric and Related Distributions of Order .'' Commun. Stat.: Th. and Meth. 19, 1291-1301, 1990.

Godbole, A. P. and Papastavnidis, G. (Eds.). Runs and Patterns in Probability: Selected Papers. New York: Kluwer, 1994.

Gordon, L.; Schilling, M. F.; and Waterman, M. S. ``An Extreme Value Theory for Long Head Runs.'' Prob. Th. and Related Fields 72, 279-287, 1986.

Goulden, I. P. and Jackson, D. M. Combinatorial Enumeration. New York: Wiley, 1983.

Mood, A. M. ``The Distribution Theory of Runs.'' Ann. Math. Statistics 11, 367-392, 1940.

Philippou, A. N. and Makri, F. S. ``Successes, Runs, and Longest Runs.'' Stat. Prob. Let. 4, 211-215, 1986.

Schilling, M. F. ``The Longest Run of Heads.'' Coll. Math. J. 21, 196-207, 1990.

Schuster, E. F. In Runs and Patterns in Probability: Selected Papers (Ed. A. P. Godbole and S. Papastavridis). Boston, MA: Kluwer, pp. 91-111, 1994.

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