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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.
1 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 CarloMonte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. Monte Carlo methods are extremely important in computational phys simulation.
Randomness is also important in other activities such as cryptographyCryptography (from Greek kryptos "hidden", and graphein "to write") is, traditionally, the study of means of converting information from its normal, comprehensible form into an incomprehensible format, rendering it unreadable without secret knowledge — th and gamblingGambling (or betting is any behaviour involving the risk of money or valuables on the outcome of a game, contest, or other event in which the outcome of that activity is partially or totally dependent upon chance. Though for many it is a form of recreatio, while pseudo-randomA pseudorandom number generator PRNG is an algorithm which generates a sequence of numbers, the elements of which are approximately independent of each other. The outputs of pseudorandom number generators are not truly random—they only approximate some of numbers are of general importance in programming and computer science.
2 Reliable sources of random numbers
2.1 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.
