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A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal die roll, or the digits of Pi (as far as we can tell) exhibit statistical randomness. Rolling dice A die (Old French de, from Latin datum something given or played [1]) is a small polyhedral object (usually a cube) suitable as a gambling device (especially for craps or sic bo). ...
Lower-case pi The mathematical constant Ï€ is a real number which is defined as the ratio of a circles circumference (Greek πεÏιφÎÏεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...
Statistical randomness does not necessarily imply "true" randomness, i.e., unpredictability. Pseudorandomness is sufficient for many uses. It has been suggested that random number be merged into this article or section. ...
A pseudorandom process is a process that appears random but is not. ...
A distinction is sometimes made between global randomness and local randomness. Most philosophical conceptions of randomness are "global" — they are based on the idea that "in the long run" a sequence would look truly random, even if certain sequences would not look random (in a "truly" random sequence of near-infinite length, for example, it is probable that there would be long sequences of nothing but zeros, though on the whole the sequence might be "random"). "Local" randomness refers to the idea that there can be minimum sequence lengths in which "random" distributions are approximated. Long stretches of the same digits, even those generated by "truly" random processes, would diminish the "local randomness" of a sample (it might only be locally random for sequences of 10,000 digits; taking sequences of less than 1,000 might not appear "random" at all, for example).
A sequence exhibiting a pattern is not thereby proved not statistically random. According to principles of Ramsey theory, sufficiently large objects must necessarily contain a given structure ("complete disorder is impossible"). Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ...
Legislation concerning gambling imposes certain standards of statistical randomness to slot machines. Gambling has had many different meanings depending on the cultural and historical context in which it is used. ...
Slot machines in the Trump Taj Mahal A slot machine (American English), poker machine (Australian English), or fruit machine (British English) is a certain type of casino game. ...
Contrast with algorithmic randomness. Chaitin-Kolmogorov randomness (also called algorithmic randomness) defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. ...
Tests
The first tests for random numbers were published by M.G. Kendall and Bernard Babington Smith in the Journal of the Royal Statistical Society in 1938. They were built on statistical tools such as Pearson's chi-square test which were developed in order to distinguish whether or not experimental phenomena matched up with their theoretical probabilities (Pearson developed his test originally by showing that a number of dice experiments by W.F.R. Weldon did not display "random" behavior). Sir Maurice George Kendall (September 6, 1907-March 29, 1983) was a prominent British statistician. ...
The Royal Statistical Society is a learned society for statistics and a professional body for statisticians in the UK. Founded in 1834 as the Statistical Society of London, it has 7200 members in the UK and the rest of the world, around 1500 of whom are professionally qualified. ...
1938 (MCMXXXVIII) was a common year starting on Saturday (link will take you to calendar). ...
Pearsons chi-square test (χ2) is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. ...
Walter Frank Raphael Weldon Walter Frank Raphael Weldon (15 March 1860 — 13 April 1906) was an English evolutionary zoologist and biometrician. ...
Kendall and Smith's original four tests were hypothesis tests, which took as their null hypothesis the idea that each number in a given random sequence had an equal chance of occurring, and that various other patterns in the data should be also distributed equiprobably. One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ...
In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ...
- The frequency test, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, 2s, 3s, etc.
- The serial test, did the same thing but for sequences of two digits at a time (00, 01, 02, etc.), comparing their observed frequencies with their hypothetical predictions were they equally distributed.
- The poker test, tested for certain sequences of five numbers at a time (aaaaa, aaaab, aaabb, etc.) based on hands in the game poker.
- The gap test, looked at the distances between 0s (00 would be a distance of 0, 010 would be a distance of 1, 02250 would be a distance of 3, etc.).
If a given sequence was able to pass all of these tests within a given degree of significance (generally 5%), then it was judged to be, in their words "locally random". Kendall and Smith differentiated "local randomness" from "true randomness" in that many sequences generated with truly random methods might not display "local randomness" to a given degree — very large sequences might contain many rows of a single digit. This might be "random" on the scale of the entire sequence, but in a smaller block it would not be "random" (it would not pass their tests), and would be useless for a number of statistical applications. This article is about the card game. ...
As random number sets became more and more common, more tests, of increasing sophistication were used. Some modern tests plot random digits as points on a three-dimensional plane, which can then be rotated to look for hidden patterns. In 1995, the statistician George Marsaglia created a set of tests known as the Diehard tests which he distributes with a CD-ROM of 5 billion pseudorandom numbers. George Marsaglia is a professor of statistics at Florida State University. ...
The Diehard tests are a battery of statistical tests for measuring the quality of a set of random numbers. ...
The CD-ROM (an abbreviation for Compact Disc Read-Only Memory (ROM)) is a non-volatile optical data storage medium using the same physical format as audio compact discs, readable by a computer with a CD-ROM drive. ...
A pseudo-random number is a number belonging to a sequence which appears to be random, but can in fact be generated by a finite computation. ...
Pseudorandom number generators require tests as exclusive verifications for their "randomness" as they are decidedly not produced by "truly random" processes, but rather by deterministic algorithms. Over the history of random number generation, many sources of numbers thought to appear "random" under testing have later been discovered to be very non-random when subjected to certain types of tests. The notion of quasi-random numbers was developed in order to circumvent some of these problems, though pseudorandom number generators are still extensively used in many applications (even ones known to be extremely "non-random"), as they are "good enough" for most applications. A pseudorandom number generator (PRNG) is an algorithm that generates a sequence of numbers, the elements of which are approximately independent of each other. ...
In mathematics, a low-discrepancy sequence is a sequence with the property that for all N, the subsequence x1, ..., xN is almost uniformly distributed (in a sense to be made precise), and x1, ..., xN+1 is almost uniformly distributed as well. ...
Other tests :
Entropy of a Bernoulli trial as a function of success probability. ...
Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...
See also Sometimes when choosing a coin (particularly for a coin flip), it may be desirable to determine if the coin is fair – that is, if the probability of obtaining a given side (commonly heads or tails) in the toss is 50%. // Posterior probability density function One way of verifying this is...
In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. ...
It has been suggested that random number be merged into this article or section. ...
It has been suggested that this article or section be merged into randomness. ...
One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ...
References - M.G. Kendall and B. Babington Smith, "Randomness and Random Sampling Numbers," Journal of the Royal Statistical Society 101:1 (1938), 147-166.
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