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References and Notes

1. Freund, John E.; Mathematical Statistics, 5th Ed.; Prentice Hall, NJ, 1992. pp185-190. A common example of a binomial random variable is the probability of getting a certain number on the roll of a die. It is fairly easy to derive the distribution function. The probability of getting say a 1 whenever a die is rolled is one chance in six, which is 1/6 or 0.1667. The probability of getting five 1's in eight rolls of a die is the probability of getting five 1's and three "non-1's" in any order. The probability of getting five 1's is (0.1667)⁵ . The probability of three non-1's is (1-0.1667)³. The probabablity of getting these results in a particular order is (0.1667)⁵ x (1-0.1667)³. If all of the die were different colors, the total number of ways you could arrange them would be 8 x 7 x .... x 2 x 1 = 8! since you can put any one of the 8 die in the first position, and for each of these any one of tne 7 remaining die in the second position, etc. Since order does not matter here, you must divide 8! by all of the duplicate patterns. The duplicate patterns are all the ways you can arrange the 1's, which is 5!, and all the ways you can arrange the non-1's, which is 3!. This shows that the number of ways to roll five 1's in eight rolls of a die is: 8! x 0.1667⁵ x (1-0.1667)³ / (5! x 3!). This is the case of a binomial distribution with the probability of a success (rolling a 1) being Q = 01667, the number of successes X=5, and the number of trials T = 8. It is easy to generalize this for any probability Q, number of successes X and number of trials T as: T! x Q^T x (1-Q)^1-T / [X! x (T-X)!]
    
2. Ibid. p. 48.
    
3. The mean m of the maximum order statistic from a random sample of size D-2 A is easily derived, but the standard deviation is non-trivial. For random samples of size 1000 from from a standard normal population m is 3.25. s appears experimentally to be approximately 0.35.
    
4. Ibid. p. 295; Central Limit Theorem.
    
5. Washburn, Del; Theomatics II, Scarborough House, Boston, 1994. pp 286-91.
    
6. Results in this table are from Theomatics Research Code.
    
7. Ibid. pp. 92-97, 616. Theomatics is characterized by a clustering phenomenon in which the number of hits at an exact accuracy of 1 (excluding hits with an accuracy of 0) is significantly higher (often twice or more) the number of hits at an exact accuracy of 2. If theomatics were random, the number of hits at any level of accuracy should be the same as any other level. The fact that they are not is appropriately cited as further evidence of the non-random nature of the theomatic phenomenon. For the purposes of approximating the probability of the general random occurence of theomatics, this fact is useful in determining an appropriate accuracy to use in testing how non-random theomatics is. Using an accuracy of 2 dampens the testing results since the results are more random at this level. An accuracy of 1 seemed preferable to an accuracy of 0 to permit more occurences of the phenomenon to study.

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