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In statistics a random number is a single observation
(outcome) of a specified random variable. Where no distribution is
specified, the continuous uniform distribution on the
interval [0,1] is usually, but not always, intended.
In an informal sense, there is some circularity in this definition as the idea of random variable itself rests on the concept of randomness. A number itself cannot be random except in the sense of how it was generated. Informally, to generate a random number means that
before it was generated, all elements of some set
were equally probable as outcomes. In particular, this means that knowledge
of earlier numbers generated by this process, or some other process, do not yield any
extra information about the next number. This is equivalent to statistical independence.
Importance of random numbers
Statistical
practice is based on statistical theory which, itself, if
founded on the concept of randomness. Many elements of statistical practice
depend on the emulation of randomness through random numbers. Where those
random numbers fall short of the conceptual ideal of randomness any subsequent
statistical analysis may suffer from bias. Elements of statistical practice that depend on randomness
include: choosing a representative sample, disguising the
protocol of a study from a participant (see randomized controlled trial) and Monte Carlo simulation.
Randomness is also important in other activities such as cryptography and gambling.
Reliable sources of random numbers
Tables of random numbers
Tables of random numbers have the desired properties no matter how chosen from the table: by row, column, diagonal or
irregularly. Originally generated by hand, they are now, more commonly, the tabulated outputs of hardware random number generators. An
important 20th century work was the RAND Corporation
million-number table. It was produced in the 1950's by an electronic simulation of a roulette wheel attached to a computer, the results of which were then carefully filtered and tested before being
used to generate the table. The RAND table was an important break-through in delivering
random numbers because such a large and carefully prepared table had never before been available.
If carefully prepared, the filtering and testing processes remove any noticeable bias or asymmetry from the hardware-generated original numbers so that such tables provide the most
'reliable' random numbers available to the casual user. But note that any published table (and in fact any previously prepared
table at all) are unusable for cryptographic purposes since the existence of the public (or private) table provides a way for an
attacker to break any cryptographic algorithm using the random numbers as an input. In short, the numbers in such tables
are not unpredictable; they can be stolen or copied by an attacker.
Hardware random-number generators
Some physical phenomena, such as thermal noise in zener diodes appear to
be truly random and can be used as the basis for hardware random number generators. However, many mechanical phenomena feature asymmetries
and biases that make their outcomes not truly random. The
many successful attempts to exploit such phenomena by gamblers, especially in
roulette and blackjack are
testimony to these effects.
There are several imaginative sources of random numbers online, most notable perhaps is LavaRand[1] which creates random numbers from images taken of a lava lamp.
Random.org[2] has a more obvious approach of listening to atmospheric noise. Details about
how they turn their input into random numbers can be found on their respective sites.
Sources that approximate random numbers
Pseudo-random numbers
Pseudo-random number generators
(PRNGs) are algorithms that can automatically create long runs (up to millions of
numbers) with good random properties but eventually the sequence repeats exactly. One of the most common PRNG is the linear congruential generator which uses the
recurrence
- Xn + 1 = aXn + b(mod
m)
to generate numbers. The maximum number of numbers the formula can produce is the modulus, m. See the article in question for
more details. Another, much earlier method of determining random number was the so-called middle square method. The method is quite simple to understand, it is however not a great
generator. You take the previous number, square it and extract the middle part of the square to use as the next number.
They are very useful in developing Monte Carlo simulations
as debugging is facilitated by the ability to run the same sequence of random
numbers again by starting from the same seed. They are also used in cryptography so long as the seed is secret. Sender and receiver can generate the same set of numbers
automatically to use as keys.
Hardware random-number generators
Many mechanical methods of generating random numbers tend to be unreliable. Hardware random number generators need much care is needed in adequate mixing and
checking randomness before use.
Testing random numbers
A variety of hypothesis tests are
used in checking random numbers including:
The null hypothesis of such tests is always that of randomness so we are principally interested in the nature of their type II errors which are typically difficult to quantify.
External Links
- [3] A big table of random
numbers
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